Elementos Simétricos e Antissimétricos sob Involuções Orientadas Generalizadas em Anéis de Grupo

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ✲ ❯❋❇❆

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ■▼

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P●▼❆❚

❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦

  

❊❧❡♠❡♥t♦s ❙✐♠étr✐❝♦s ❡ ❆♥t✐ss✐♠étr✐❝♦s s♦❜

■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s ❡♠ ❆♥é✐s ❞❡

●r✉♣♦

  

❊❞✇❛r❞ ▲❛♥❞✐ ❚♦♥✉❝❝✐

  ❙❛❧✈❛❞♦r✲❇❛❤✐❛ ✷✺ ❞❡ ❆❣♦st♦ ❞❡ ✷✵✶✼

  

❊❧❡♠❡♥t♦s ❙✐♠étr✐❝♦s ❡ ❆♥t✐ss✐♠étr✐❝♦s s♦❜

■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s ❡♠ ❆♥é✐s ❞❡

●r✉♣♦

  

❊❞✇❛r❞ ▲❛♥❞✐ ❚♦♥✉❝❝✐

  ❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❯❋❇❆✴❯❋❆▲ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✱ ❛♣r♦✈❛❞❛ ❡♠ ✷✺ ❞❡ ❆❣♦st♦ ❞❡ ✷✵✶✼✳ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❚❤✐❡rr② ❈♦rrê❛ P❡t✐t ▲♦❜ã♦✳

  ❙❛❧✈❛❞♦r✲❇❛❤✐❛ ✷✺ ❞❡ ❆❣♦st♦ ❞❡ ✷✵✶✼

  ❚♦♥✉❝❝✐✱ ❊❞✇❛r❞ ▲❛♥❞✐✳ ❊❧❡♠❡♥t♦s ❙✐♠étr✐❝♦s ❡ ❆♥t✐ss✐♠étr✐❝♦s s♦❜ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s

●❡♥❡r❛❧✐③❛❞❛s ❡♠ ❆♥é✐s ❞❡ ●r✉♣♦✴ ❊❞✇❛r❞ ▲❛♥❞✐ ❚♦♥✉❝❝✐✳ ✕ ❙❛❧✈❛❞♦r✿

❯❋❇❆✱ ✷✵✶✼✳

  ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❚❤✐❡rr② ❈♦rrê❛ P❡t✐t ▲♦❜ã♦✳ ❉✐ss❡rt❛çã♦ ✭❞♦✉t♦r❛❞♦✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ✷✵✶✼✳ ❘❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✳

  ✶✳ ❆♥é✐s ✭➪❧❣❡❜r❛✮✳ ✷✳ ❆♥é✐s ❞❡ ●r✉♣♦✳ ✸✳ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s✳ ■✳ P❡t✐t

▲♦❜ã♦✱ ❚❤✐❡rr② ❈♦rrê❛✳ ■■✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡

▼❛t❡♠át✐❝❛✳ ■■■✳ ❚ít✉❧♦✳ ❈❉❯ ✿ ✺✶✷✳✺✺✷✳✼

  

❊❧❡♠❡♥t♦s ❙✐♠étr✐❝♦s ❡ ❆♥t✐ss✐♠étr✐❝♦s s♦❜

■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s ❡♠ ❆♥é✐s ❞❡

●r✉♣♦

  

❊❞✇❛r❞ ▲❛♥❞✐ ❚♦♥✉❝❝✐

  ❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❯❋❇❆✴❯❋❆▲ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✱ ❛♣r♦✈❛❞❛ ❡♠ ✷✺ ❞❡ ❆❣♦st♦ ❞❡ ✷✵✶✼✳

  ❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

  Pr♦❢✳ ❉r✳ ❚❤✐❡rr② ❈♦rrê❛ P❡t✐t ▲♦❜ã♦ ✭❖r✐❡♥t❛❞♦r✮ ❯❋❇❆

  ❛

  Pr♦❢ ✳ ❉r❛✳ ▼❛♥✉❡❧❛ ❞❛ ❙✐❧✈❛ ❙♦✉③❛ ❯❋❇❆

  ❛

  Pr♦❢ ✳ ❉r❛✳ ❈❛r♠❡❧❛ ❙✐❝❛ ❯❋❇❆

  

  Pr♦❢ ✳ ❉r❛✳ ❆♥❛ ❈r✐st✐♥❛ ❱✐❡✐r❛ ❯❋▼●

  

  Pr♦❢ ✳ ❉r✳ ❖s♥❡❧ ❇r♦❝❤❡ ❈r✐st♦

  ❆♦s ♠❡✉s ♣❛✐s✱ ❈❧á✉❞✐♦ ❡ ❘✐t❛✱ ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r❛ ❏❛❝q✉❡❧✐♥❡ ❡ ♠❡✉s ✐r♠ã♦s ❏♦ã♦ P❛✉❧♦ ❡ ❈❛r♦❧✐♥❡✳

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  Pr✐♠❡✐r♦✱ ❛❣r❛❞❡ç♦ ✐♠❡♥s❛♠❡♥t❡ à ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣♦r t♦❞♦ ❛♣♦✐♦ ♠♦r❛❧ ❡ ✜♥❛♥✲ ❝❡✐r♦✱ ♣♦✐s s❡♠ ❡st❡ t❡♥❤♦ ❝❡rt❡③❛ q✉❡ ♥✉♥❝❛ ❛❧❝❛♥ç❛r✐❛ ♦ ♥í✈❡❧ ✐♥t❡❧❡❝t✉❛❧✱ ❛❝❛❞ê♠✐❝♦ ❡ ♣r♦✜ss✐♦♥❛❧ ♥♦ q✉❛❧ ♠❡ ❡♥❝♦♥tr♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡ç♦ ♠✉✐t♦ ❛♦s ♠❡✉s ♣❛✐s ❈❧á✉❞✐♦ ❡ ❘✐t❛ ♣♦r t♦❞♦ ❛♠♦r✱ ❝❛r✐♥❤♦✱ ✐♥❝❡♥t✐✈♦ ❡ ❡❞✉❝❛çã♦ ♠♦r❛❧✱ ♣♦✐s ❝❛❞❛ ✉♠ ❞❡ss❡s ❞❡t❛❧❤❡s ❛❥✉❞♦✉ ❛ ❝♦♥str✉✐r ♠✐♥❤❛ ♣❡rs♦♥❛❧✐❞❛❞❡ ❡ ❛ ♣❡ss♦❛ q✉❡ s♦✉✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦s ♠❡✉s ✐r♠ã♦s✱ ❏♦ã♦ P❛✉❧♦ ❡ ❈❛r♦❧✐♥❡✱ ❡ ❛♦s ♠❡✉s ♣r✐♠♦s ♣❡❧♦s q✉❛✐s ♣♦ss✉♦ t❛♥t♦ ❛❢❡t♦✱ ♣♦r t♦❞♦ ♦ ❝❛r✐♥❤♦✱ ❛♠✐③❛❞❡ ❡ ❞✐✈❡rsã♦ q✉❡ ♠❡ ♣r♦♣♦r❝✐♦♥❛r❛♠ ❞❡s❞❡ ❛ ♠✐♥❤❛ ✐♥❢â♥❝✐❛ ❛té ♦ ♣r❡s❡♥t❡ ♠♦♠❡♥t♦✳

  ❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r❛ ❏❛❝q✉❡❧✐♥❡ ♣♦r t♦❞♦ ❝❛r✐♥❤♦✱ ❛♠♦r✱ ❝♦♥✜❛♥ç❛✱ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦ ♠❡ ❛❥✉❞❛♥❞♦ ❛ ♣❛ss❛r ♣♦r t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s q✉❡ ❛♣❛r❡❝❡r❛♠ ♥❡st❛ ❥♦r♥❛❞❛✱ ✐♥❝❡♥t✐✈❛♥❞♦ ❡ ❛♣♦✐❛♥❞♦ ❛s ❞❡❝✐sõ❡s ✐♠♣♦rt❛♥t❡s q✉❡ t✐✈❡ q✉❡ t♦♠❛r ❛♣ós ✐♥✐❝✐❛r ♠✐♥❤❛ ❝❛rr❡✐r❛ ❛❝❛❞ê♠✐❝❛✱ ❛❧é♠ ❞❡ s❡r ✉♠❛ ♣❡ss♦❛ ❡①❝❡♣❝✐♦♥❛❧ ❝♦♠ q✉❡♠ ♣♦ss♦ ❞✐✈✐❞✐r ♠❡✉s ❞❡s❡❥♦s ❡ ❣♦st♦s✳

  ❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s✱ ❡①✲❝♦❧❡❣❛s ❞❡ ❣r❛❞✉❛çã♦✱ ♠❡str❛❞♦ ❡ ❞♦✉t♦r❛❞♦ ♣❡❧♦s ❜♦♥s ♠♦♠❡♥t♦s q✉❡ ♣❛ss❛♠♦s ❡st✉❞❛♥❞♦ ❡ ♣♦r ❛❥✉❞❛r❡♠ ❛ ♠❡ ❞✐✈❡rt✐r ❡ ❞✐str❛✐r ♥♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝❛♥s♦✳ ❆❣r❛❞❡ç♦ ❡♠ ❡s♣❡❝✐❛❧ às ♠✐♥❤❛s ✐r♠ãs ❆❧❡❥❛♥❞r❛ ❡ ❊❧❡♥ ♣♦r t♦❞♦ ♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❞✉r❛♥t❡ t♦❞❛ ❛ ❥♦r♥❛❞❛✳

  ❆❣r❛❞❡ç♦ ♠✉✐t♦ t❛♠❜é♠ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❚❤✐❡rr② P❡t✐t ▲♦❜ã♦✱ t❛♥t♦ ♣❡❧❛ s✉❛ ❞✐s♣♦s✐çã♦✱ ❞❡❞✐❝❛çã♦ ❡ ♣r❡st❛t✐✈✐❞❛❞❡ q✉❛♥t♦ ♣❡❧♦ s❡✉ ❝♦♥st❛♥t❡ ✐♥❝❡♥t✐✈♦ ❡ ♣r♦✜ss✐♦♥❛✲ ❧✐s♠♦ ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ❥♦r♥❛❞❛ ❞❛ ♣ós✲❣r❛❞✉❛çã♦✳

  ❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ▼❛♥✉❡❧❛ ❞❛ ❙✐❧✈❛ ❙♦✉③❛✱ ❈❛r♠❡❧❛ ❙✐❝❛✱ ❆♥❛ ❈r✐st✐♥❛ ❱✐❡✐r❛ ❡ ❖s♥❡❧ ❇r♦❝❤❡ ❈r✐st♦ ♣♦r ♣❛rt✐❝✐♣❛r❡♠ ❞❛ ❝♦♠✐ssã♦ ❥✉❧❣❛❞♦r❛ ❞❛ ♠✐♥❤❛ t❡s❡ ❡ ♠❡ ❞❛r❡♠ ❛ ❣r❛♥❞❡ ❤♦♥r❛ ❞❡ tê✲❧♦s ❝♦♠♦ ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❞❡ ♠✐♥❤❛ ❞❡❢❡s❛✳

  ❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❈❊✲❯❊❙❇ ❡ ■▼✲❯❋❇❆ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❡❢❡t✐✈❛♠❡♥t❡ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❝♦♠♦ ♠❛t❡♠át✐❝♦✳ ❆❣r❛❞❡ç♦ ❛✐♥❞❛ ♠❛✐s ❛♦s q✉❡ ❝♦♥✲ tr✐❜✉ír❛♠ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♥ã♦ s♦♠❡♥t❡ ❝♦♠♦ ♠❛t❡♠át✐❝♦✱ ♠❛s t❛♠❜é♠ ❝♦♠♦ s❡r ❤✉♠❛♥♦✳ ❆❣r❛❞❡❝✐♠❡♥t♦s ❡s♣❡❝✐❛✐s ❛♦s ♣r♦❢❡ss♦r❡s ❆❞❡❧③✐t♦✱ ❆❝✐♦❧②✱ ❈❧❛✉❞✐♥❡✐✱ ❈❧ê♥✐❛✱ ❉é❜♦r❛✱ ❊r✐❞❛♥✱ ❋❧❛✉❧❧❡s✱ ❏ú❧✐♦✱ ▼ár❝✐♦✱ ❘❡❣✐♥❛❧❞♦✱ ❚â♥✐❛✱ ❉✐❡❣♦ ❡ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❆✉❣✉st♦✱ ♣♦r s❡✉s ❡①❡♠♣❧♦s ❝♦♠♦ ♣r♦✜ss✐♦♥❛✐s ❡ ót✐♠♦s ♣r♦❢❡ss♦r❡s✳

  ❋✐♥❛❧♠❡♥t❡✱ ❛❣r❛❞❡ç♦ à ❈❆P❊❙ ❡ à ❋❆P❊❙❇ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ ❝♦♥❝❡❞✐❞♦ ❛

  ✏❙♦♥❤♦ q✉❡ s❡ s♦♥❤❛ só é só ✉♠ s♦♥❤♦ q✉❡ s❡ s♦♥❤❛ só✳ ▼❛s s♦♥❤♦ q✉❡ s❡ s♦♥❤❛ ❥✉♥t♦ é r❡❛❧✐❞❛❞❡✳✑

  ✕❏♦❤♥ ▲❡♥♦♥✱ tr❛❞✉③✐❞♦ ♣♦r ❘❛✉❧ ❙❡✐①❛s

  ❘❡s✉♠♦

  ❙❡❥❛ RG ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦ ❞❡ ✉♠ ❣r✉♣♦ G s♦❜r❡ ✉♠ ❛♥❡❧ R t❛❧ q✉❡ char(R) 6= 2✳ ❆ ♣❛rt✐r ❞❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ σ : G → {±1}✱ ❝❤❛♠❛❞♦ ❞❡ ♦r✐❡♥t❛çã♦ ❡♠ G✱ ❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗✱ ♣♦❞❡♠♦s ♠✉♥✐r RG ❝♦♠ ✉♠❛ ✐♥✈♦❧✉çã♦ ❞❡ ❛♥é✐s σ∗✱ ❝❤❛♠❛❞❛ ❞❡ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛✱ ❛ ♣❛rt✐r ❞❛ ❡①t❡♥sã♦ ❧✐♥❡❛r ❞❡ ∗ t♦r❝✐❞❛ ♣❡❧❛ ♦r✐❡♥t❛çã♦ σ✳

  ➱ ♥♦tá✈❡❧ ❛ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ♣❡sq✉✐s❛ r❡❧❛❝✐♦♥❛♥❞♦ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❡ ✐♥✲ ✈♦❧✉çõ❡s✱ s❡♥❞♦ q✉❡✱ ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❡♠ RG✱ ♣❛rt❡ ❞❡ss❛ é ❢♦❝❛❞❛ ♥♦ ❡st✉❞♦

  = = α ❞❛ r❡❧❛çã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ✈❡r✐✜❝❛❞❛ ❡♠ RG − ∗ + {α ∈ RG : α } ♦✉

  = ❡♠ RG {α ∈ RG : α −α} ❡ ❛s ❡str✉t✉r❛s ❜❛s❡ ❞❡st❡ ❛♥❡❧✱ ♥♦ ❝❛s♦ R ❡ G✳ + ❖s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s q✉❡ ❡①✐❜❡♠ ❡st❛s r❡❧❛çõ❡s sã♦ ❛❧❝❛♥ç❛❞♦s q✉❛♥❞♦ RG ♦✉ RG ✱ s♦❜ ✐♥✈♦❧✉çõ❡s ❞❛❞❛s ♣❡❧❛ ❡①t❡♥sã♦ ❧✐♥❡❛r ❞❡ ∗ ❡♠ RG ♦✉ ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s σ

  ∗✱ ✈❡r✐✜❝❛♠ ❝♦♠✉t❛t✐✈✐❞❛❞❡✱ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ♦✉ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ▲✐❡✳ ◆❡st❡ s❡♥t✐❞♦✱ ❜✉s❝❛♠♦s ♥♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡st✉❞❛r ♦s ❝❛s♦s ♥ã♦ ❡①♣❧♦r❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛✱ ❛s✲ s✐♠ ❝♦♠♦ ❣❡♥❡r❛❧✐③❛r ♦s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ♣❛r❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛✱ ♦♥❞❡ s✉♣♦♠♦s ❛ ♦r✐❡♥t❛çã♦ σ ❛❣♦r❛ s❡♥❞♦ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ G ❡♠ U(R)✳

  ❈❧❛ss✐✜❝❛r❡♠♦s ❞❡ ❢♦r♠❛ ♣❛r❝✐❛❧ q✉❛♥❞♦ RG ✱ s♦❜ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛✱ é + ❝♦♠✉t❛t✐✈♦✱ s❡♥❞♦ ❡st❡ ♦ ú❧t✐♠♦ ❝❛s♦ ❛ s❡r tr❛t❛❞♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❡①♣♦st♦ ❛❝✐♠❛✱ ❡✱ ❛❧é♠ ❞✐st♦✱ ❞❡s❝r❡✈❡r❡♠♦s ❞❡ ❢♦r♠❛ ✐♥t❡❣r❛❧ q✉❛♥❞♦ ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ é ✈❡r✐✜❝❛❞❛ ❡♠ + RG

  ♦✉ RG ❛❣♦r❛ s♦❜ ❛ ót✐❝❛ ❞❛s ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s✳ P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❆♥é✐s ❞❡ ●r✉♣♦s❀ ■♥✈♦❧✉çõ❡s❀ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s❀ ❈♦♠✉t❛t✐✈✐❞❛❞❡❀ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡✳

  ❆❜str❛❝t

  ▲❡t RG ❜❡ ❛ ❣r♦✉♣ r✐♥❣ ♦❢ ❛ ❣r♦✉♣ G ♦✈❡r ❛ r✐♥❣ R ✇✐t❤ charR 6= 2✳ ❋r♦♠ ❛ ❤♦♠♦♠♦r✜s♠ σ : G → {±1}✱ ❝❛❧❧❡❞ ♦r✐❡♥t❛t✐♦♥ ✐♥ G✱ ❛♥❞ ❛♥ ✐♥✈♦❧✉t✐♦♥ ∗✱ ✇❡ ❝❛♥ ❛ss✐❣♥ RG

  ✇✐t❤ ❛ r✐♥❣ ✐♥✈♦❧✉t✐♦♥ σ∗✱ ❝❛❧❧❡❞ ♦r✐❡♥t❡❞ ✐♥✈♦❧✉t✐♦♥✱ ❜② t❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦❢ ∗ t✇✐st❡❞ ❜② t❤❡ ♦r✐❡♥t❛t✐♦♥ σ✳ ❆ r❡♠❛r❦❛❜❧❡ ❛♠♦✉♥t ♦❢ r❡s❡❛r❝❤ ❤❛s ❜❡ ❞♦♥❡ ❝♦♥♥❡❝t✐♥❣ ♣♦❧②♥♦♠✐❛❧ ✐❞❡♥t✐t✐❡s

  ✇✐t❤ ✐♥✈♦❧✉t✐♦♥s✳ ■♥ t❤✐s s❡♥s❡✱ ♣❛rt ♦❢ t❤❡ r❡s❡❛r❝❤✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ✐♥✈♦❧✉t✐♦♥ ♦♥ ❘●✱ + = = α

  ✐s ❢♦❝✉s❡❞ ♦♥ t❤❡ st✉❞② ♦❢ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❛♥ ✐❞❡♥t✐t② ✐♥ RG {α ∈ RG : α } − ∗ =

  ♦r ✐♥ RG {α ∈ RG : α −α} ❛♥❞ t❤❡ ❜❛s✐❝ str✉❝t✉r❡s ♦❢ t❤❡ r✐♥❣✱ ♥❛♠❡❧②✱ ❘ ❛♥❞ ●✳ + ❚❤❡ ♠❛✐♥ r❡s✉❧ts ❝♦♥❝❡r♥✐♥❣ t❤❡s❡ r❡❧❛t✐♦♥s ❛r❡ ♦❜t❛✐♥❡❞ ✇❤❡♥ RG ♦r RG ✱

  ✉♥❞❡r ✐♥✈♦❧✉t✐♦♥s ❣✐✈❡♥ ❜② t❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦❢ ∗ ✐♥ RG ♦✉ t❤❡ ♦r✐❡♥t❡❞ ✐♥✈♦❧✉t✐♦♥s σ∗✱ s❛t✐✜❡s ❝♦♠♠✉t❛t✐✈✐t② ♦r ❛♥t✐❝♦♠♠✉t❛t✐✈✐t②✱ ♦r ❡✈❡♥ s♦♠❡ ▲✐❡ ♣r♦♣❡rt②✳ ■♥ t❤✐s ✇❛②✱ ✇❡ ✐♥t❡♥❞ t♦ ❛❞❞r❡ss ❝❛s❡s ♥♦t ②❡t ❡①♣❧♦r❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ❛s ✇❡❧❧ ❛s

  ❡①t❡♥❞✐♥❣ ❦♥♦✇♥ r❡s✉❧ts t♦ ❛ ❣❡♥❡r❛❧✐③❡❞ ♦r✐❡♥t❡❞ ✐♥✈♦❧✉t✐♦♥✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♦r✐❡♥t❛t✐♦♥ σ

  ❛s ❛ ❤♦♠♦♠♦r✜s♠ ♦❢ G ✐♥t♦ U(R)✳ + ❲❡ s❤❛❧ ♣❛rt✐❛❧❧② ❝❧❛ss✐❢② t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ RG ✱ ✉♥❞❡r ❛♥ ♦r✐❡♥t❡❞ ✐♥✈♦❧✉t✐♦♥✱

  ✐s ❝♦♠♠✉t❛t✐✈❡✱ s✉❝❤ ✐s t❤❡ ❧❛st ❝❛s❡ t♦ ❜❡ tr❡❛t❡❞ ✐♥ t❤❡ s❡♥s❡ ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✳ ❇❡s✐❞❡s✱ + ✇❡ s❤❛❧❧ ❢✉❧❧② ❞❡s❝r✐❜❡ t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ ❛♥t✐❝♦♠♠✉t❛t✐✈✐t② ✐s ✈❡r✐✜❡❞ ✐♥ RG ♦r RG ✉♥❞❡r t❤❡ ❧✐❣❤t ♦❢ ❣❡♥❡r❛❧✐③❡❞ ♦r✐❡♥t❡❞ ✐♥✈♦❧✉t✐♦♥s✳ ❑❡②✇♦r❞s✿ ●r♦✉♣ ❘✐♥❣s❀ ■♥✈♦❧✉t✐♦♥s❀ ❖r✐❡♥t❡❞ ■♥✈♦❧✉t✐♦♥s❀ ●❡♥❡r❛❧✐③❡❞ ❖r✐❡♥t❡❞ ■♥✈♦✲ ❧✉t✐♦♥s❀ ❈♦♠♠✉t❛t✐✈✐t②❀ ❆♥t✐❝♦♠♠✉t❛t✐✈✐t②✳

  ❮♥❞✐❝❡

  ■♥tr♦❞✉çã♦ ✶

  ✶ Pr❡❧✐♠✐♥❛r❡s ✼

  ✶✳✶ ❆♥é✐s ❞❡ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ■♥✈♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

  ✶✳✷✳✶ ■♥✈♦❧✉çõ❡s ❡♠ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✷ ■♥✈♦❧✉çõ❡s ❡♠ ❆♥é✐s ❞❡ ●r✉♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

  ✶✳✸ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✹ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

  ✷ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❆♥t✐ss✐♠étr✐❝♦s ✶✾ ✸ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❙✐♠étr✐❝♦s

  ✸✷ ✹ ❈♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❆♥t✐ss✐♠étr✐❝♦s

  ✺✶ ✹✳✶ R 2 6= {0} ♦✉ (G ∩ Z(G))\N 6= ∅ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

  ✹✳✶✳✶ G 2 6⊂ G ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✹✳✶✳✷ G 2 ⊂ G ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

  ✺ ❈♦♥❝❧✉sõ❡s ✻✻

  ✺✳✶ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✺✳✷ ❚r❛❜❛❧❤♦s ❋✉t✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

  ❘❡❢❡rê♥❝✐❛s ✻✽

  ■♥tr♦❞✉çã♦

  ❆ ❚❡♦r✐❛ ❞♦s ❆♥é✐s ❞❡ ●r✉♣♦s ❝✉♠♣r❡ ✉♠ ♣❛♣❡❧ ❞❡ ❞❡st❛q✉❡ ❞❡♥tr♦ ❞❛ ♠❛t❡♠á✲ t✐❝❛✱ ✈✐st♦ q✉❡ ❡ss❛ ❢✉♥❝✐♦♥❛ ❝♦♠♦ ✉♠ ♣♦♥t♦ ❡♠ q✉❡ t❡♠♦s ✉♠❛ ✈✐❛ ❞❡ ♠ã♦ ❞✉♣❧❛ ❧✐❣❛♥❞♦ ❛s ♠❛✐s ❞✐✈❡rs❛s s✉❜ár❡❛s ❞❛ á❧❣❡❜r❛✱ ♥❛ q✉❛❧ ❝♦♥❤❡❝❡♥❞♦ r❡s✉❧t❛❞♦s ❛❝❡r❝❛ ❞❡ ❣r✉♣♦s✱ ❛♥é✐s✱ ♠ó❞✉❧♦s ❡ á❧❣❡❜r❛s✱ ♣♦❞❡♠♦s ❛✈❛♥ç❛r ❝♦♠ ❛ t❡♦r✐❛ ❡♠ q✉❡stã♦ ❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥❢♦r♠❛çõ❡s q✉❡ ♣♦ss✉í♠♦s ❞❡ ✉♠ ❞❛❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦✱ ♣♦❞❡♠♦s ❡①tr❛✐r ♣r♦♣r✐❡❞❛❞❡s ❜❛st❛♥t❡ ❢♦rt❡s ❞❡ss❛s ❡str✉t✉r❛s✱ ❛❧é♠ ❞❡ t♦♠á✲❧❛s ❝♦♠♦ ✉♠ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ❡ ♠♦t✐✈❛çã♦ ♣❛r❛ ❛ ❜✉s❝❛ ❞❡ ❛♥á❧♦❣♦s ♣❛r❛ ❛s ❚❡♦r✐❛s ❞❡ ❆♥é✐s ❡ ▼ó❞✉❧♦s✳

  ❯♠ ❡①❡♠♣❧♦ ❝❧❛r♦ ❞❡ ❝♦♠♦ ❢✉♥❝✐♦♥❛ ❡ss❛ tr♦❝❛ ❞❡ ✐♥❢♦r♠❛çõ❡s ♣♦❞❡ s❡r ✈✐st❛ ♥♦s r❡s✉❧t❛❞♦s ❛❜♦r❞❛❞♦s ♥❡ss❡ tr❛❜❛❧❤♦✳ ❉❛❞♦ ✉♠ ❛♥❡❧ R ❝♦♠ ✐♥✈♦❧✉çã♦ ∗✱ ❝❤❛♠❛♠♦s ❞❡ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s s✐♠étr✐❝♦s +

  ❡ ❛♥t✐ss✐♠étr✐❝♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛♦s ❝♦♥❥✉♥t♦s ∗ − ∗ R := = r := =

  {r ∈ R : r } ❡ R {r ∈ R : r −r} ; ✉t✐❧✐③❛♠♦s t❛♠❜é♠ ❛ ♥♦t❛çã♦ R ❡ R ∗ q✉❛♥❞♦ é ❝♦♥✈❡♥✐❡♥t❡ ❡s♣❡❝✐✜❝❛r ❛ ✐♥✈♦❧✉çã♦ ❡♠ t❡❧❛✳ ❖ ✐♥t❡r❡ss❡ ❡♠ ❡st✉❞❛r t❛✐s ❝♦♥❥✉♥t♦s t♦r♥♦✉✲s❡ ❡✈✐❞❡♥t❡ ❛♣ós ❆♠✐ts✉r t❡r ♣r♦✈❛❞♦

  ❡♠ ✉♠ ❛rt✐❣♦ ❝❧áss✐❝♦✱ ❬❆✻✾❪✱ q✉❡ s❡ R ♦✉ R s❛t✐s❢❛③ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧✱ + ❡♥tã♦ R t❛♠❜é♠ s❛t✐s❢❛③ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧✳

  ■♥❢❡❧✐③♠❡♥t❡✱ ♥♦ ❝♦♥t❡①t♦ ❞♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✱ ❛ ✐❞❡♥t✐❞❛❞❡ q✉❡ R ✈❡r✐✜❝❛✱ ♥ã♦ é✱ + ❡♠ ❣❡r❛❧✱ ❛ ♠❡s♠❛ ✈❡r✐✜❝❛❞❛ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ ♣r❡✈✐❛♠❡♥t❡ ❝♦♥❤❡❝✐❞❛ ❡♠ R ♦✉ R ✳ ❈❛s♦ s❛t✐s❢❛ç❛ ❛ ♠❡s♠❛ ✐❞❡♥t✐❞❛❞❡ ❡♥❝♦♥tr❛❞❛ ❡♠ R ✭♦✉ R ✮✱ ❞✐③❡♠♦s q✉❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❢♦✐

  • + R ❧❡✈❛♥t❛❞❛ ❞♦s s✐♠étr✐❝♦s ✭♦✉ ❞♦s ❛♥t✐ss✐♠étr✐❝♦s✮ ♣❛r❛ R✳ ❉❡ss❛ ❢♦r♠❛✱ s✉r❣❡ ❛ ♣❡r❣✉♥t❛ ♣❛r❛ q✉❛✐s ✐❞❡♥t✐❞❛❞❡s ♦ ❧❡✈❛♥t❛♠❡♥t♦ r❡❢❡r✐❞♦ ❛❝✐♠❛ é ❣❛r❛♥t✐❞♦✳

  ❚♦♠❛♥❞♦ R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡ ❡ G ✉♠ ❣r✉♣♦ q✉❛❧q✉❡r✱ ❞❡✜♥✐♠♦s RG

  ❝♦♠♦ ♦ R✲♠ó❞✉❧♦ ❧✐✈r❡♠❡♥t❡ ❣❡r❛❞♦ ♣♦r G✳ ▼✉♥✐♥❞♦ RG ❞❡ ✉♠ ♣r♦❞✉t♦ ❛ ♣❛rt✐r ❞❛s ♦♣❡r❛çõ❡s ❞❡ R ❡ G✱ r❡s♣❡✐t❛♥❞♦ ❛ ❞✐str✐❜✉t✐✈✐❞❛❞❡✱ t❡♠♦s q✉❡ RG ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛ ♥❛t✉r❛❧ ❞❡ ❛♥❡❧✱ ♦ q✉❛❧ ❝❤❛♠❛♠♦s ❞❡ ❛♥❡❧ ❞❡ ❣r✉♣♦✳

  ❆ ♣❛rt✐r ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❞❡ ❣r✉♣♦s ❡♠ G✱ é ♣♦ssí✈❡❧ ❡st❡♥❞❡r R✲❧✐♥❡❛r♠❡♥t❡ ❡ss❛ ✐♥✈♦❧✉çã♦ ♣❛r❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ❞❡ ❛♥é✐s ♥♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG✱ ♥❡ss❡ ❝❛s♦ t❛♠❜é♠

  P

  α x x ❞❡♥♦t❛❞❛ ♣♦r ∗✱ ♦✉ s❡❥❛✱ s❡ α = x ✱ ❡♥tã♦ ∗ ∗ ∈G X ! ∗ X

  α = α x x = α x x , x ∈G x ∈G ❝❤❛♠❛❞❛ ❞❡ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ ❞❡ G ❡♠ RG✳ ◆♦t❡ q✉❡ ❛ ❛♣❧✐❝❛çã♦ q✉❡ ❧❡✈❛ ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❣r✉♣♦ ❡♠ s❡✉ ✐♥✈❡rs♦ é ✉♠❛ ✐♥✈♦❧✉çã♦ ❞❡ ❣r✉♣♦s❀ ❝❤❛♠❛♠♦s ❞❡ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛ ❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG ❛ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ ♣♦r ❡ss❛ ❛♥t❡r✐♦r✳ ❖❜✈✐❛♠❡♥t❡✱ ❞❛❞❛ ✉♠❛ + ✐♥✈♦❧✉çã♦ ∗ ✐♥❞✉③✐❞❛ ❡♠ ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦s s✉❜❝♦♥❥✉♥t♦s RG ❡ RG

  ✳ ❉❛❞♦ ✉♠ ❛♥❡❧ R✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ [r, s] = rs−sr ❡✱ ♣♦r ✐♥❞✉çã♦✱ 1 2 1 2 3 i [r , r , . . . , r n ] = [[r , r ], r , . . . , r n ],

  ♣❛r❛ t♦❞♦ r, s, r ∈ R✳ ❚r✐✈✐❛❧♠❡♥t❡ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A ⊂ R é ❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ [a, b] = 0, ∀a, b ∈ A❀ ♥❡ss❡ s❡♥t✐❞♦✱ ❢❛③❡♥❞♦ ✉♠❛ ❡①t❡♥sã♦ ❞❡ss❡ ❝♦♥❝❡✐t♦✱ ♣♦❞❡♠♦s ❡st✉❞❛r ♦s s✉❜❝♦♥❥✉♥t♦s A ⊂ R t❛✐s q✉❡ 1

  [a , . . . , a n ] = 0, ♦✉

  [a, b, . . . , b ] = 0, | {z } n i , b

✈❡③❡s

♣❛r❛ t♦❞♦ a, a ∈ A✳ ❉✐③❡♠♦s q✉❡ ✉♠ s✉❜❝♦❥✉♥t♦ q✉❡ ✈❡r✐✜❝❛ ❛ ♣r✐♠❡✐r❛ ✐❞❡♥t✐❞❛❞❡✱ ♣❛r❛ ❛❧❣✉♠ n✱ é ▲✐❡ ♥✐❧♣♦t❡♥t❡✱ ❡ ▲✐❡ n✲❊♥❣❡❧✱ s❡ ❡st❡ ✈❡r✐✜❝❛ ❛ s❡❣✉♥❞❛✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ ❈♦♠✉t❛t✐✈✐❞❛❞❡ ⇒ ▲✐❡ ♥✐❧♣♦tê♥❝✐❛ ⇒ ▲✐❡ n✲❊♥❣❡❧.

  ❊♠ ❬●❙✾✸❪✱ ●✐❛♠❜r✉♥♦ ❡ ❙❡❤❣❛❧ ♠♦str❛r❛♠ q✉❡ s❡ K é ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡ 2 ❡ G ✉♠ ❣r✉♣♦ q✉❡ ♥ã♦ ♣♦ss✉✐ 2✲❡❧❡♠❡♥t♦s✱ ❡♥tã♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ▲✐❡ + ♥✐❧♣♦tê♥❝✐❛ ♣♦❞❡ s❡r ❧❡✈❛♥t❛❞❛ ❞❡ KG ❡ KG ✱ ♥♦ ❝❛s♦ ❞❡ ∗ s❡r ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✳ ❊♠ ❬▲✾✾✱ ▲✵✵❪✱ ▲❡❡ ❡st❡♥❞❡✉ ❡ss❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❣r✉♣♦s ♠❛✐♦r❡s ❡ ♣❛r❛ ♦ ❝❛s♦ ▲✐❡ n✲❊♥❣❡❧✱ ❛❧é♠ ❞❡ ❡st✉❞❛r q✉❛♥❞♦ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ♥ã♦ é ♣♦ssí✈❡❧✳ P♦st❡r✐♦r♠❡♥t❡✱ ❡♠ ❬●P❙✵✾✱ ▲❙❙✵✾❪✱ ♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛ ❢♦r❛♠ tr❛❜❛❧❤❛❞♦s ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ q✉❛❧q✉❡r✳ +

  ◆♦t❡ q✉❡ ♥♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ RG ❡ RG sã♦ ❝♦♠✉t❛t✐✈♦s✱ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❡ss❛ ✐❞❡♥t✐❞❛❞❡ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ❣r✉♣♦ G s❡r ❛❜❡❧✐❛♥♦✳ ❉❡ss❛ ❢♦r♠❛ ❡st✉❞❛r ❛ ❝♦♠✉t❛✲ t✐✈✐❞❛❞❡ ❞♦s s✐♠étr✐❝♦s s❡ r❡s✉♠❡ ❛♦ ❡st✉❞♦ ❞❛ ❡str✉t✉r❛ ❞♦ ❣r✉♣♦ G✳ ❆ ❝❛r❛❝t❡r✐③❛çã♦ ♥❡ss❡s ❝❛s♦s s❡ ❡♥❝♦♥tr❛ ❡♠ ❬❏❘✵✻✱ ❇❏P❘✵✾❪✳

  ❯♠❛ ♦✉tr❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ♥❛ ár❡❛ s✉r❣❡ ❛ ♣❛rt✐r ❞♦ Pr♦❞✉t♦ ❞❡ ❏♦r❞❛♥✱ ❞❛❞♦ ♣♦r r ◦ s = rs + sr, ∀r, s ∈ R✳ ❉✐③❡♠♦s q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A ⊂ R é

  ✹ + ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡ ♦ Pr♦❞✉t♦ ❞❡ ❏♦r❞❛♥ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ s♦❜r❡ ❡ss❡ ❝♦♥❥✉♥t♦✱ ♦✉ s❡❥❛✱ r

  ◦s = 0, ∀r, s ∈ A✳ ◆ã♦ é ❞✐✜❝í❧ ✈❡r✐✜❝❛r q✉❡ ♥♦ ❝❛s♦ ❞❡ RG ♦✉ RG s❡r ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❡ss❛ ✐❞❡♥t✐❞❛❞❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ G s❡r ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❡ charR = 2✱ ♣♦rt❛♥t♦✱ ❛ss✐♠ ❝♦♠♦ ♥♦ ❝❛s♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡✱ ♦ ❡st✉❞♦ é ❢❡✐t♦ s♦❜r❡ ❛ ❡str✉t✉r❛ ❞♦ ❣r✉♣♦ G✱ ❡ ❡st❡ ❢♦✐ ❝♦♠♣❧❡t❛♠❡♥t❡ ❝❛r❛❝t❡r✐③❛❞♦ ❡♠ ❬●P✶✸❛❪✳

  ❉❛❞♦ ❛s r❡❢❡rê♥❝✐❛s ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ♦s + ❝♦♥❥✉♥t♦s RG ❡ RG ❡ ❛s ✐❞❡♥t✐❞❛❞❡s ❝✐t❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✱ ♣♦✉❝♦ t❡♠✲s❡ ❛ ❛❝r❡s❝❡♥t❛r✳ ❉✐t♦ ✐st♦✱ s✉r❣❡♠ ❛s ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s σ∗ ❡♠ ❛♥é✐s ❞❡ ❣r✉♣♦s✱ q✉❡ sã♦ ❞❛❞❛s ♣❡❧❛ ❡①t❡♥sã♦ ❧✐♥❡❛r ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❡♠ G ❛❝r❡s❝✐❞❛ ❞❡ ✉♠❛ t♦rsã♦ ♣r♦✈❡♥✐❡♥t❡ ❞❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ σ : G → {±1}✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ σ∗ : RG → RG ❝♦♠ σ ∗ ∗ X ! ∗ σ X α = α x x = σ(x)α x x . x x ∈G ∈G

  ❚❛✐s ✐♥✈♦❧✉çõ❡s ❢♦r❛♠ ✐♥tr♦❞✉③✐❞❛s ♣♦r ◆♦✈✐❦♦✈✱ ❡♠ ❬◆✼✵❪✱ ♥♦ ❝♦♥t❡①t♦ ❞❛ K✲t❡♦r✐❛✳ ◆❛t✉r❛❧♠❡♥t❡ ♣❛r❛ q✉❡ ❡st❛ ✐♥✈♦❧✉çã♦ s❡❥❛ ❞✐st✐♥t❛ ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♥ã♦ ♦r✐❡♥t❛❞❛✱ é ♥❡❝❡ssár✐♦ q✉❡ char(R) 6= 2✱ ♦ q✉❛❧ s❡rá ❤✐♣ót❡s❡ ❛ss✉♠✐❞❛ ❡♠ t♦❞♦ ♦ tr❛❜❛❧❤♦✳ = : n

  ❚♦♠❛♥❞♦ N := kerσ✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ s❡ N {n ∈ N} ⊂ N✱ ❛ r❡str✐çã♦ ❞❡ σ∗ ❛♦ s✉❜❛♥❡❧ RN ⊂ RG ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ RN✳ P♦r ♦✉tr♦ ❧❛❞♦✱ + ❛❞♠✐t✐♥❞♦ q✉❡ RG ♦✉ RG s❛t✐s❢❛ç❛ ❛❧❣✉♠❛ ❞❛s ✐❞❡♥t✐❞❛❞❡s r❡❢❡r✐❞❛s ❛❝✐♠❛✱ t❡♠♦s q✉❡ + RN

  ♦✉ RN t❛♠❜é♠ ♦ ❢❛③✱ ❞❡ss❛ ❢♦r♠❛✱ ♦s r❡s✉❧t❛❞♦s ❥á ❝✐t❛❞♦s ❢♦r♥❡❝❡♠ ✐♥❢♦r♠❛çõ❡s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ❛ ❡str✉t✉r❛ ❞❡ N ❡✱ ❛ ♣❛rt✐r ❞✐ss♦✱ ♣♦❞❡♠♦s ❞❡s❝r❡✈❡r ♦ ❣r✉♣♦ G✳ ❚❛✐s té❝♥✐❝❛s ✜❝❛rã♦ ♠❛✐s ❝❧❛r❛s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳ ❙✉r♣r❡❡♥❞❡♥t❡♠❡♥t❡✱ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❝❛s♦s✱ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❝♦♥t✐♥✉❛ s❡♥❞♦ ❣❛r❛♥t✐❞♦ ❝♦♠ ❛s ♠❡s♠❛s ❤✐♣ót❡s❡s ❞♦s ❝❛s♦s ♥ã♦ ♦r✐❡♥t❛❞♦s❀ ♥♦ ❝❛s♦ ❞♦ ♥ã♦ ❧❡✈❛♥t❛♠❡♥t♦✱ t❛♠❜é♠ é ♣♦ssí✈❡❧ ✈❡r✐✜❝❛r q✉❡✱ σ ❡①❝❡t♦ ❡♠ ❛❧❣✉♠❛s ✐♥❞❡♥t✐❞❛❞❡s✱ ♦s ❣r✉♣♦s t❛✐s q✉❡ (RG) ♦✉ (RG) σ ∗ s❛t✐s❢❛③❡♠ ❛❧❣✉♠❛ ❞❛s ✐❞❡♥t✐❞❛❞❡s ❝✐t❛❞❛s ♣❡rt❡♥❝❡♠ ❛ ♠❡s♠❛ ❝❧❛ss❡ ❞♦ ❝❛s♦ ♥ã♦ ♦r✐❡♥t❛❞♦✳ ❚❛✐s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬❈P✶✷✱ ●P✶✸❜✱ ●P✶✹❪✱ ♣♦r ❡①❡♠♣❧♦✳

  ❊st❡♥❞❡♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛✱ ❝❤❡❣❛♠♦s ✜♥❛❧♠❡♥t❡ às ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s✱ ♦❜t✐❞❛s ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ s✉❜st✐t✉✐r ♦ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❛ ♦r✐✲ ❡♥t❛çã♦ σ ❞❡ {±1} ♣♦r U(R)✱ ♦ ❣r✉♣♦ ❞❛s ✉♥✐❞❛❞❡s ❞♦ ❛♥❡❧ R✳ ◆❡ss❡ ❝❛s♦✱ ♣❛r❛ q✉❡ ❛ ❛♣❧✐❝❛çã♦ σ∗ s❡❥❛ ❞❡ ❢❛t♦ ✉♠❛ ✐♥✈♦❧✉çã♦✱ ❛ ❝♦♥❞✐çã♦ N ⊂ N s❡ tr❛❞✉③ ♥❡ss❡ ❝❛s♦ ♣♦r xx

  ∈ N✳ ❊♠ ❬❱✶✸❪✱ ❱✐❧❧❛ ❡st✉❞♦✉ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ▲✐❡ s♦❜ ❛ ót✐❝❛ ❞❡ss❛ ♥♦✈❛ ✐♥✈♦❧✉çã♦✱

  ❣❡♥❡r❛❧✐③❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ❡♠ ❬❈P✶✷❪✱ ♠♦str❛♥❞♦✱ ❞❡ ❢♦r♠❛ s✐♠✐❧✐❛r ❛♦s ❝❛s♦s ❛♥t❡r✐♦r❡s✱ q✉❡ ♦ ❣r✉♣♦ G ♥ã♦ ♣♦ss✉✐ ❡❧❡♠❡♥t♦s ❞❡ ♦r❞❡♠ 2 é ✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ + ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ ▲✐❡ ✈❡r✐✜❝❛❞❛s ❡♠ KG ❡ KG ✱ ♥♦ ❝❛s♦ ❞❡ K s❡r ✉♠ ❝♦r♣♦ t❛❧ q✉❡ charK 6= 2✳

  ◆❡ss❡ tr❛❜❛❧❤♦ ❝♦♥t✐♥✉❛r❡♠♦s ❡st❡♥❞❡♥❞♦ r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ❞❛ t❡♦r✐❛✱ ❡st✉✲

  ✺ ❞❛♥❞♦ ❛❧❣✉♥s ❝❛s♦s ❛✐♥❞❛ ♥ã♦ ❛❜♦r❞❛❞♦s✱ r❡❧❛❝✐♦♥❛❞♦s à ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✲ + ✈✐❞❛❞❡ ❞♦s ❝♦♥❥✉♥t♦s RG ❡ RG ✳

  ◆♦ ❈❛♣ít✉❧♦ ✷✱ ❝❛r❛❝t❡r✐③❛r❡♠♦s ♦ ❣r✉♣♦ G✱ ❛ss✐♠ ❝♦♠♦ ✐r❡♠♦s ❢♦r♥❡❝❡r ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s r❡❧❛t✐✈❛s ❛♦ ❛♥❡❧ R ♣❛r❛ q✉❡ ♦ ❝♦♥❥✉♥t♦ RG s❡❥❛ ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ♥♦ ❝❛s♦ ❞❡ σ s❡r ✉♠❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛✱ ❡st❡♥❞❡♥❞♦ ❛ss✐♠ ♦s r❡s✉❧t❛❞♦s ❝♦♥t✐❞♦s ❡♠ ❬●P✶✸❜❪✳

  ◆❡ss❡ ❝❛s♦✱ ♦❜s❡r✈❛r❡♠♦s q✉❡ ❛ ❝❧❛ss❡ ❞❡ ❣r✉♣♦s ♣❛r❛ q✉❡ RG s❡❥❛ ❛♥t✐❝♦♠✉t❛t✐✈♦ ♣❡r✲ ♠❛♥❡❝❡ ❛ ♠❡s♠❛ ❞♦s ❝❛s♦s ❡♠ q✉❡ σ é ❛ ♦r✐❡♥t❛çã♦ ❝❧áss✐❝❛ ✭σ ≡ {±1}✮ ♦✉ ✉♠❛ ♦r✐❡♥t❛çã♦ tr✐✈✐❛❧✳

  ◆♦ ❈❛♣ít✉❧♦ ✸✱ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ ❛♥á❧♦❣♦ ❛♦ ❞♦ ❈❛♣ít✉❧♦ ✷✱ ♣♦ré♠ ❛❞♠✐t✐♥❞♦ + ❛❣♦r❛ q✉❡ ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ é ✈❡r✐✜❝❛❞❛ ❡♠ RG ✳ ❊ss❡ ❝❛s♦ ✈❡♠ ❛ t❡r ✉♠ ❞❡st❛q✉❡ ♥❡ss❡ tr❛❜❛❧❤♦✱ ✈✐st♦ q✉❡✱ ❛s ❝❧❛ss❡s ❞❡ ❣r✉♣♦s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ❡ss❛ ✐❞❡♥t✐❞❛❞❡ ♥❡ss❡ ❝♦♥❥✉♥t♦ sã♦ ❛s ♠❡s♠❛s ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ♦r✐❡♥t❛çã♦ σ ❝❧áss✐❝❛ ♦✉ tr✐✈✐❛❧✱ ❛♦ ♣❛ss♦ q✉❡ ❛♦ ❢❛③❡r♠♦s ❛ ❡①t❡♥sã♦ ♣❛r❛ ❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛✱ ✉♠❛ ♥♦✈❛ ❝❧❛ss❡ ❞❡ ❣r✉♣♦s ❛♣❛r❡❝❡ ❝♦♠♦ ♣♦ss✐❜✐❧✐❞❛❞❡✳ ❖✉tr♦ ❢❛t♦ ✐♥t❡r❡ss❛♥t❡ é q✉❡✱ ♣❛r❛ ❝♦♥❝❧✉✐r ♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✷ ❡ ❡♠ ❬❱✶✸❪✱ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ s✉❜❣r✉♣♦ N✱ ❡st✉❞❛❞♦s q✉❛♥❞♦ ❛ ♦r✐❡♥t❛çã♦ é tr✐✈✐❛❧ ❡♠ ❬●P✶✸❛✱ ▲✾✾✱ ▲✵✵❪✱ ❢♦✐ s✉✜❝✐❡♥t❡ ♣❛r❛ t❛❧ ❝❛r❛❝t❡r✐③❛çã♦✱ ♣♦rt❛♥t♦✱ ♦s r❡s✉❧t❛❞♦s ❛ss♦❝✐❛❞♦s ❛ ❝❛❞❛ ✉♠ ❞♦s ❝❛s♦s ♣❛r❛ ❛s ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s ♥ã♦ ❢♦r❛♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ ✉t✐❧✐③❛❞♦s✱ ❡ ♣♦❞❡r✐❛♠ s❡r tr❛t❛❞♦s ❝♦♠♦ ❝♦r♦❧ár✐♦s ❞❡ss❛ ❡①t❡♥sã♦❀ s✉r♣r❡❡♥❞❡♥t❡♠❡♥t❡ t❛❧ ❢❛t♦ ♥ã♦ s❡ r❡♣❡t❡ ♥♦ t❡♠❛ ❛❜♦r❞❛❞♦ ♥❡st❡ ❝❛♣ít✉❧♦✱ ❞❡ss❛ ❢♦r♠❛✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠ ♥♦✈♦ s✉❜❣r✉♣♦ C ⊂ G ❞❛❞♦ ♣♦r

  C = {x ∈ G : σ(x) = ±1} , q✉❡ ❡ss❡♥❝✐❛❧♠❡♥t❡ é ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ s♦❜ ♦s q✉❛✐s ❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐✲

  ③❛❞❛ s❡ ❝♦♠♣♦rt❛ ❝♦♠♦ ❛ ♦r✐❡♥t❛çã♦ ❝❧áss✐❝❛✱ s❡♥❞♦ ❛ss✐♠✱ ♦ ❡st✉❞♦ ❞❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❡❧❡♠❡♥t♦s s✐♠étr✐❝♦s s♦❜ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛ s❡r✐❛ ✐♠♣♦ss✐❜✐❧✐t❛❞♦ s❡♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ❝❛s♦ σ ≡ {±1}✱ ❡♥❝♦♥tr❛❞♦ ❡♠ ❬●P✶✹❪✳

  P♦r ✜♠✱ ♥♦ ❈❛♣ít✉❧♦ ✹✱ ✐♥✐❝✐❛r❡♠♦s ♦ ❡st✉❞♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❛♥t✐ss✐♠étr✐✲ ❝♦s ♣❛r❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ✉s✉❛❧✱ s❡♥❞♦ q✉❡ ❡st❡ é ♦ ú❧t✐♠♦ ❝❛s♦ ❛ s❡r tr❛t❛❞♦ ♣❛r❛ ♦ ❢❡❝❤❛♠❡♥t♦ ❞♦ q✉❛❞r♦ ❞❡ss❡ t✐♣♦ ❞❡ ✐♥✈♦❧✉çã♦ ✭♥❛ ✈❡r❞❛❞❡✱ ♣❛r❛ ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✱ ❥á ❡①✐st❡ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ q✉❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬❇❏❘✵✼❪✮✳ ❊st✉❞❛r ❡ss❛ ♣❛rt✐❝✉❧❛r ✐♥✈♦❧✉çã♦ é ❥✉st✐✜❝❛❞❛ ❞❡✈✐❞♦ ❛♦ ❣r❛✉ ♠❛✐♦r ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡ss❡ ❝❛s♦✱ ✈✐st♦ q✉❡ ❞❡s❞❡ ❛ ❛❜♦r❞❛❣❡♠ ♥ã♦ ♦r✐❡♥t❛❞❛✱ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❥á ❛♣r❡s❡♥t❛ ✉♠❛ ❝❧❛ss❡ ♠❛✐♦r ❞❡ ❣r✉♣♦s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❛♥❡❧ R✱ ♦ q✉❡ t♦r♥❛ ❛✐♥❞❛ ♠❛✐s ❝♦♠♣❧❡①♦ ♦ ❡st✉❞♦ ❞♦s ❝❛s♦s ♦r✐❡♥t❛❞♦s❀ ❛❞❡♠❛✐s✱ ❛ss✐♠ ❝♦♠♦ ♦ q✉❡ ❢♦✐ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❈❛♣ít✉❧♦ ✸✱ é ♣♦ssí✈❡❧ q✉❡ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦ ❞❡ss❡ ❝❛s♦ s❡❥❛ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ♦r✐❡♥t❛çã♦ ❣❡✲ = = x ♥❡r❛❧✐③❛❞❛✳ ❉❡✈✐❞♦ ❛ ❞✐✜❝✉❧❞❛❞❡ ❥✉st✐✜❝❛❞❛ ❛❝✐♠❛✱ ❞❡♥♦t❛♥❞♦ G {x ∈ G : x }✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠❛ r❡s♣♦st❛ ♣❛r❛ ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡ (G ∩ Z(G))\N 6= ∅ ♦✉ ♦

  ✻ ❛♥❡❧ R ♣♦ss✉✐ ❡❧❡♠❡♥t♦s ❞❡ t♦rsã♦ 2✱ ♦✉ s❡❥❛✱ ♦ ❝♦♥❥✉♥t♦ 2 R :=

  {r ∈ R : 2r = 0} , é ❞✐❢❡r❡♥t❡ ❞❡ {0}✳

  ❈❛♣ít✉❧♦ ✶ Pr❡❧✐♠✐♥❛r❡s

  ◆❡st❡ ❝❛♣ít✉❧♦✱ ✐r❡♠♦s ❛❜♦r❞❛r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ tr❛❜❛❧❤♦✳

  ✶✳✶ ❆♥é✐s ❞❡ ●r✉♣♦s

  ❏á ✈✐♠♦s ❝♦♠♦ ♦s ❛♥é✐s ❞❡ ❣r✉♣♦ ❢✉♥❝✐♦♥❛♠ ❝♦♠♦ ♣♦♥t♦ ❞❡ ❡♥❝♦♥tr♦ ❞❛s ❞✐✈❡rs❛s t❡♦r✐❛s ❛❧❣é❜r✐❝❛s✳ ■r❡♠♦s ❛❣♦r❛ ❛♣r❡s❡♥t❛r✱ ❝♦♠ ♠❛✐s ♣r❡❝✐sã♦✱ ❡ss❛ ❡str✉t✉r❛✳ ❊♠❜♦r❛ ♥ã♦ s❡❥❛ ✉♠❛ ❡①✐❣ê♥❝✐❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ ❞❡✜♥✐çã♦✱ ❛❞♠✐t✐r❡♠♦s s❡♠♣r❡ q✉❡ ♦ ❛♥❡❧ R t♦♠❛❞♦ ♣♦r ❜❛s❡ ❞❡ ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG✱ s❡❥❛ ❝♦♠✉t❛t✐✈♦ ❡ ❝♦♠ ✉♥✐❞❛❞❡

  1 R ∈ R✳

  ❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛♠ G ✉♠ ❣r✉♣♦ ❡ R ✉♠ ❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡✳ ❉❡♥♦t❡ ♣♦r RG ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s ❢♦r♠❛✐s ❞❛ ❢♦r♠❛ X x x x α = a x x, g ∈G

  ♦♥❞❡ a ∈ R ❡ {a } x é q✉❛s❡ ♥✉❧❛✱ ♦✉ s❡❥❛✱ a 6= 0 ❛♣❡♥❛s ♣❛r❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ∈G ❞❡ í♥❞✐❝❡s✳ ❖ ❝♦♥❥✉♥t♦ (RG, +, ·) ❞♦t❛❞♦ ❞❛s ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡ ♣r♦❞✉t♦ ❞❡✜♥✐❞♦ ❞❛ ❢♦r♠❛ ❛ s❡❣✉✐r é ✉♠ ❛♥❡❧✱ ❝❤❛♠❛❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ ❞❡ G s♦❜r❡ R✿ X ! ! X X X x x x ∈G ∈G ∈G ! ! + a x x b x x = (a x + b x )x; X X a x x b x x = (a x b y )(xy). x x x,y ∈G ∈G ∈G · RG = 1 R

  1 G ◆♦t❡ q✉❡ RG é ✉♠ ❛♥❡❧ ❝♦♠ ✐❞❡♥t✐❞❛❞❡✱ ♦♥❞❡ 1 ✱ q✉❡✱ ❞❡ ❛❣♦r❛ ❡♠

  ❞✐❛♥t❡✱ ❞❡♥♦t❛r❡♠♦s s✐♠♣❧❡s♠❡♥t❡ ♣♦r 1✳

  X ✽ a x x

  ❉❛❞♦ α = ∈ RG✱ ❝❤❛♠❛♠♦s ❞❡ s✉♣♦rt❡ ❞❡ α✱ supp(α)✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s x ∈G ❡❧❡♠❡♥t♦s x ∈ G q✉❡ ❛♣❛r❡❝❡♠ ♥❛ ❝♦♠♣♦s✐çã♦ ❞❡ α ❞❡ ❢♦r♠❛ ♥ã♦ tr✐✈✐❛❧✱ ♦✉ s❡❥❛✱ supp(α) = x

  {x ∈ G : α 6= 0} . P♦❞❡♠♦s t❛♠❜é♠ ❞❡✜♥✐r ✉♠ ♣r♦❞✉t♦ ♣♦r ❡s❝❛❧❛r❡s ❞♦ ❛♥❡❧ R ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛ X ! X r a x x = ra x x,

  · ∀r ∈ R, x x ∈G ∈G ❡ ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛♠♦s q✉❡ RG é ✉♠ R✲♠ó❞✉❧♦✳ ❆❞❡♠❛✐s✱ s❡ R é ❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ RG é ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ R✳

  ❉✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦✱ ❡ ♣❡❧♦ ❢❛t♦ ❞❡ R s❡r ❝♦♠✉t❛t✐✈♦✱ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r q✉❡ RG s❡rá ❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ G ❢♦r ❛❜❡❧✐❛♥♦✳ ∞ , x , x , x , x , . . . −2 −1 1 2 ❊①❡♠♣❧♦ ✶✳✷✳ ❙❡❥❛♠ G = C ≃ {. . . , x } ❡ R = R✱ ♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❚❡♠♦s q✉❡ RG = RC é ✐s♦♠♦r❢♦ ❛♦ ❛♥❡❧ ❞♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ▲❛✉r❡♥t✳ G

  ❯t✐❧✐③❛♥❞♦ ♦ ♠♦♥♦♠♦r✜s♠♦ ❞❡ ✐♥❝❧✉sã♦ i : R → RG ❞❡✜♥✐❞♦ ♣♦r i(r) 7→ r1 ✱ t❡♠♦s ♥❛t✉r❛❧♠❡♥t❡ q✉❡ RG ❝♦♥té♠ ✉♠ s✉❜❛♥❡❧ ✐s♦♠♦r❢♦ ❛ R✱ ♦ q✉❛❧ ❢r❡q✉❡♥t❡♠❡♥t❡ tr❛t❛r❡♠♦s ❝♦♠♦ ♦ ♣ró♣r✐♦ R❀ ❛♥❛❧♦❣❛♠❡♥t❡✱ ❝♦♠♦ R ♣♦ss✉✐ ✉♥✐❞❛❞❡✱ ❞❛❞♦ ❛ ✐♥❝❧✉sã♦ x R x,

  7→ 1 ∀x ∈ G✱ t❡♠♦s q✉❡ ♦ ❣r✉♣♦ ❞❛s ✉♥✐❞❛❞❡s ❞♦ ❛♥❡❧ RG✱ U(RG)✱ ♣♦ss✉✐ ✉♠ s✉❜❣r✉♣♦ ✐s♦♠♦r❢♦ ❛♦ ❣r✉♣♦ G✱ q✉❡ t❛♠❜é♠ s❡rá ✐❞❡♥t✐✜❝❛❞♦ ♣♦r G✳

  ✶✳✷ ■♥✈♦❧✉çõ❡s

  ■r❡♠♦s ❛♣r❡s❡♥t❛r ♥❡st❛ s❡çã♦ ❛ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ✉t✐❧✐③❛❞❛ ♥❡ss❡ tr❛❜❛❧❤♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ❛♥é✐s ❞❡ ❣r✉♣♦s✳

  ✶✳✷✳✶ ■♥✈♦❧✉çõ❡s ❡♠ ●r✉♣♦s

  ❉❡✜♥✐çã♦ ✶✳✸✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ∗ : G → G✱ ❡♠ ✉♠ ❣r✉♣♦ G✱ é ✉♠❛ ✐♥✈♦❧✉çã♦ ❞❡ ❣r✉♣♦s s❡ ✈❡r✐✜❝❛ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ ∗ ∗ ∗

  = y x ✭✐✮ (xy) ✱ ∗ ∗

  ) = x ✭✐✐✮ (x ✱

  ♣❛r❛ t♦❞♦ x, y ∈ G✳ ◆❡ss❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ G é ✉♠ ❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ∗✳ −1 ❊①❡♠♣❧♦ ✶✳✹✳ ❆ ❛♣❧✐❝❛çã♦ x 7→ x ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ q✉❛❧q✉❡r ❣r✉♣♦ G✱ ❝❤❛♠❛❞❛ ❞❡ ■♥✈♦❧✉çã♦ ❈❧áss✐❝❛✳

  ✾ ❊①❡♠♣❧♦ ✶✳✺✳ ❆ tr❛♥s♣♦s✐çã♦ ❞❡ ♠❛tr✐③❡s ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ❞❡ ❣r✉♣♦s ♥♦ ❣r✉♣♦ ❣❡r❛❧ ❧✐♥❡❛r GL(n, K)✱ s❡♥❞♦ K ✉♠ ❝♦r♣♦✳ ❊①❡♠♣❧♦ ✶✳✻✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ∗ ❡ H ≤ G✳ ❙❡ h H

  ∈ H, ∀h ∈ H✱ ❡♥tã♦ é ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ H✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ tr❛♥s♣♦s✐çã♦ é ✉♠❛ ✐♥✈♦❧✉çã♦ ♥♦ ❣r✉♣♦

  ∗| H =

  {M ∈ GL(n, K) : det(M) = ±1}✳ n = = y = 1, yx = x y n −1 2 ❊①❡♠♣❧♦ ✶✳✼✳ ❙❡❥❛ D hx, y : x i ♦ ❣r✉♣♦ ❞✐❡❞r❛❧ ❞❡ 2n ❡❧❡♠❡♥✲ n n ∗ −1 ∗

  = x , y = t♦s✳ ❆ ❛♣❧✐❝❛çã♦ ∗ : D → D ❞❡✜♥✐❞❛ ♣❡❧❛ ❡①t❡♥sã♦ ❛♦ ♣r♦❞✉t♦ ❛ ♣❛rt✐r ❞❡ x xyx

  ✱ é ✉♠❛ ✐♥✈♦❧✉çã♦ ♥❡ss❡ ❣r✉♣♦✳ y −1 −1 −1 ❉✉r❛♥t❡ t♦❞♦ ♦ tr❛❜❛❧❤♦✱ ❞❡♥♦t❛r❡♠♦s ❛s ❝♦♥❥✉❣❛çõ❡s ❡ ❝♦♠✉t❛❞♦r❡s ❡♠ G ♣♦r x = y xy y xy, ❡ (x, y) = x ∀x, y ∈ G✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ 8

  ❊①❡♠♣❧♦ ✶✳✽✳ ❙❡❥❛ ♦ Q ❣r✉♣♦ ❞♦s q✉❛tér♥✐♦s ❞❡ ♦r❞❡♠ 8✱ ❝✉❥❛ ♣r❡s❡♥t❛çã♦ é 8 4 2 2 y −1 Q = = 1, x = y , x = x 8 hx, y; x i.

  ❆ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛ ❡♠ Q é ❞❛❞❛ ❡①♣❧✐❝✐t❛♠❡♥t❡ ♣♦r ( 2 g , s❡ g ∈ {1, x } g = 2 x g , 2 8 2 ❝❛s♦ ❝♦♥trár✐♦✳ ) = Q = 8

  ◆♦t❡ q✉❡ {1, x } = Z(Q hx i✳ ❖❜s❡r✈❡ ♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ q✉❡ ❛ ✐♥✈♦❧✉çã♦ é ❞❛❞❛ ♣r❡❝✐s❛♠❡♥t❡ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ 8

  ♣♦r ❡❧❡♠❡♥t♦s ❞♦ s✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦ ❞❡ Q ✳ ❱❡r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡ q✉❡ ✐♥✈♦❧✉çõ❡s ❞❡ss❡ t✐♣♦ sã♦ r❡❝♦rr❡♥t❡s ♥❛ ❧✐♥❤❛ ❞❡ ♣❡sq✉✐s❛ ❛❜♦r❞❛❞❛ ♥❡ss❡ t❡①t♦✳

  ✶

  ❉❡✜♥✐çã♦ ✶✳✾✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❣r✉♣♦ ♥ã♦ ❛❜❡❧✐❛♥♦ G é ✉♠ ▲❈✲❣r✉♣♦ ✱ s❡ ♣❛r❛ t♦❞♦ x, y ∈ G t❛✐s q✉❡ xy = yx✱ ❡♥tã♦ x, y ♦✉ xy ∈ Z(G)✳

  ❯♠❛ ❝❧❛ss❡ ❜❛st❛♥t❡ ✐♥t❡r❡ss❛♥t❡ ❞❡ ❣r✉♣♦s ❝♦♠ ✐♥✈♦❧✉çã♦ s✉r❣❡ q✉❛♥❞♦ ❛❞♠✐t✐✲ = 2

  ♠♦s q✉❡ ✉♠ ▲❈✲❣r✉♣♦ G ♣♦ss✉✐ ✉♠ s✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦ ❞❡ ♦r❞❡♠ 2✱ ♦✉ s❡❥❛ G hsi ≃ C ✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ G ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s✳ ❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ❝❧❛ss✐✜❝❛ ❡ss❛ ❝❧❛ss❡ ❞❡ ❣r✉♣♦s✱ ❝❛r❛❝t❡r✐③❛♥❞♦✲♦s ♣♦r s❡r❡♠ ♠✉♥✐❞♦s ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♠✉✐t♦ ♣❛rt✐❝✉❧❛r ❞♦ t✐♣♦ ❝✐t❛❞♦ ♥♦ ❊①❡♠♣❧♦ ✶✳✽✳ ❚❡♦r❡♠❛ ✶✳✶✵ ✭❚❡♦r❡♠❛s ■■■✳✸✳✸ ❡ ■■■✳✸✳✻✱ ❬●❏P✾✻❪✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ♥ã♦ ❛❜❡❧✐❛♥♦✳ ❆s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

  ✭✐✮ G é ✉♠ ▲❈✲❣r✉♣♦ ❝♦♠ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s✳

  ❉♦ ✐♥❣❧ês ▲✐♠✐t❡❞ ❈♦♠♠✉t❛t✐✈✐t②✳

  −1 ∗ ✶✵ yx ✭✐✐✮ G ♣♦ss✉✐ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡ x ∈ {y, y } , ∀x, y ∈ G✱ q✉❡ ♥❡ss❡

  ❝❛s♦ é ❞❛❞❛ ♣r❡❝✐s❛♠❡♥t❡ ♣♦r ( x , s❡ x ∈ Z(G) x = sx , 2 2 s❡ x /∈ Z(G).

  ✭✐✐✐✮ G/Z(G) ≃ C × C ✳ ❉❡✜♥✐çã♦ ✶✳✶✶✳ ❖s ▲❈✲❣r✉♣♦s ❝♦♠ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s ♠✉♥✐❞♦ ❞❛ ✐♥✈♦✲

  ✷

  ❧✉çã♦ ∗ ❛♣r❡s❡♥t❛❞❛ ♥♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r sã♦ ❝❤❛♠❛❞♦s ❞❡ ❙▲❈✲❣r✉♣♦s ✳ ❆ ✐♥✈♦❧✉çã♦ ∗ é ❝❤❛♠❛❞❛ ❞❡ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ❞♦s ❙▲❈✲❣r✉♣♦s✳ 8

  ❊①❡♠♣❧♦ ✶✳✶✷✳ Q ♠✉♥✐❞♦ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦ ∗ ❞♦ ❊①❡♠♣❧♦ ✶✳✽ é ✉♠ ❙▲❈✲❣r✉♣♦✳ ◆♦t❡ 8 q✉❡ ♥❡ss❡ ❝❛s♦ ❛ ✐♥✈♦❧✉çã♦ ❞♦s ❙▲❈✲❣r✉♣♦s ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛ ❡♠ Q ✳ 4 ❊①❡♠♣❧♦ ✶✳✶✸✳ ❉❛❞♦ ❛ ♣r❡s❡♥t❛çã♦ ❞❡ D ♥♦ ❊①❡♠♣❧♦ ✶✳✼✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ❡st❡ 2

  ❣r✉♣♦ é ✉♠ ▲❈✲❣r✉♣♦ ❝♦♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ x ✱ ♣♦ré♠ ♥ã♦ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ D 4

  ♠✉♥✐❞♦ ❞❛ ✐♥✈♦❧✉çã♦ ∗ ❞❡ss❡ ♠❡s♠♦ ❡①❡♠♣❧♦ s❡❥❛ ✉♠ ❙▲❈✲❣r✉♣♦✱ ❥á q✉❡ ❡ss❛ ♥ã♦ é ❞♦ t✐♣♦ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❡st❡ ♠❡s♠♦ t❡♦r❡♠❛ ❛✜r♠❛ q✉❡ ❛ ❛♣❧✐❝❛çã♦ ( 2 w , ∗ } s❡ w ∈ {1, x w = 2 2 x w , 4 4 s❡ w /∈ {1, x } ,

  é ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ D ❀ ❛♦ ♠✉♥✐r D ❝♦♠ ❡ss❛ ✐♥✈♦❧✉çã♦✱ ❞✐③❡♠♦s q✉❡ ❡st❡ é ✉♠ ❙▲❈✲ ❣r✉♣♦✳ ❊①❡♠♣❧♦ ✶✳✶✹✳ ❆ ♣r♦♣r✐❡❞❛❞❡ ❞❡ s❡r ✉♠ ▲❈✲❣r✉♣♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ♣♦ss✉✐r ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧✳ ❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡ ♦ ❣r✉♣♦ 4 2 G = x 1 , x 2 , x 3 : x = (x , x ) = ((x , x ), x ) = 1 i i ♣❛r❛ i, j, k ❞✐st✐♥t♦s j i j k 1 , x 2 ), (x 2 , x 3 ) 1 , x 3 )

  ♣♦ss✉✐ três ❝♦♠✉t❛❞♦r❡s ♥ã♦ tr✐✈✐❛✐s (x ❡ (x ✱ ❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ G/Z(G) ♣♦ss✉✐ ❡①♣♦❡♥t❡ 2✱ ♦ q✉❡ ❣❛r❛♥t❡ q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s ♥ã♦ ❝❡♥tr❛✐s é ❝❡♥tr❛❧✱ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ▲❈✲❣r✉♣♦ é ✈❡r✜❝❛❞❛✳ ◆♦t❡ q✉❡ ❞❡✈✐❞♦ à q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠✉t❛❞♦r❡s✱ ❡st❡ ♥ã♦ s❡ ❡♥❝❛✐①❛ ♥❛ ❝❧❛ss❡ ❞♦s ❙▲❈✲❣r✉♣♦s✳ 1 2 1 2 2

  , x , x , x 3 ❖❜s❡r✈❡ q✉❡ ♥❡st❡ ❝❛s♦✱ H = hx i ❡ I = hx i sã♦ ❙▲❈✲❣r✉♣♦s ❞❡ ♦r❞❡♠ 1 2 32 , x )

  ❡ 64✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s❡♥❞♦ s = (x ♦ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ ❞❡ H ❡ I✳

  ✶✳✷✳✷ ■♥✈♦❧✉çõ❡s ❡♠ ❆♥é✐s ❞❡ ●r✉♣♦

  ❉❡✜♥✐çã♦ ✶✳✶✺✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ∗ : R → R✱ ❡♠ R é ✉♠ ❛♥❡❧✱ é ✉♠❛ ✐♥✈♦❧✉çã♦ ❞❡ ❛♥é✐s s❡ ✈❡r✐✜❝❛ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

  ∗ ∗ ∗ ✶✶

  = s r ✭✐✮ (rs) ✱ ∗ ∗ ∗

  = r + s ✭✐✐✮ (r + s) ✱ ∗ ∗

  ) = r ✭✐✐✐✮ (r ✱

  ♣❛r❛ t♦❞♦ r, s ∈ R✳ ◆❡ss❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ R é ✉♠ ❛♥❡❧ ❝♦♠ ✐♥✈♦❧✉çã♦ ∗✳ n (R) ❊①❡♠♣❧♦ ✶✳✶✻✳ ❆ tr❛♥s♣♦s✐çã♦ ❞❡ ♠❛tr✐③❡s ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♥♦ ❛♥❡❧ M ❞❛s ♠❛tr✐③❡s n × n ❝♦♠ ❡♥tr❛❞❛s ❡♠ ✉♠ ❛♥❡❧ R✳ ❊①❡♠♣❧♦ ✶✳✶✼✳ ❆ ❝♦♥❥✉❣❛çã♦ ❝♦♠♣❧❡①❛ é ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ C✳

  ❉❛s ♣♦ssí✈❡✐s ✐♥✈♦❧✉çõ❡s ❛ s❡r❡♠ ❞❡✜♥✐❞❛s ❡♠ ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG✱ ❞❡st❛❝❛♠✲s❡ ❛s ✐♥✈♦❧✉çõ❡s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❞♦ ❣r✉♣♦ G✳ ❊①❡♠♣❧♦ ✶✳✶✽ ✭■♥✈♦❧✉çã♦ ■♥❞✉③✐❞❛ ❡♠ ❆♥é✐s ❞❡ ●r✉♣♦s✮✳ ❙❡❥❛ RG ♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ ❞❡ G s♦❜r❡ R ❡ ∗ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ G✳ ❆ ❛♣❧✐❝❛çã♦✱ t❛♠❜é♠ ❞❡♥♦t❛❞❛ ♣♦r ∗✱ ∗ : RG → RG✱ ❞❡✜♥✐❞❛ ❛ ♣❛rt✐r ❞❛ ❡①t❡♥sã♦ ❧✐♥❡❛r X ! ∗ X x x ∈G ∈G α x x = α x x ,

  é ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ RG✱ ❝❤❛♠❛❞❛ ❞❡ ■♥✈♦❧✉çã♦ ■♥❞✉③✐❞❛ ❞❡ G ❡♠ RG✳ ❈♦♠♦ t♦❞♦ ❣r✉♣♦ é ♠✉♥✐❞♦ ❞❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♣♦❞❡♠♦s ❢❛③❡r ❛

  ❡①t❡♥sã♦ ❧✐♥❡❛r ❞❡✜♥✐❞❛ ❛❝✐♠❛✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ t♦❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG s❡❥❛ ♥❛t✉r❛❧♠❡♥t❡ ✉♠ ♣♦rt❛❞♦r ❞❡ss❛ ✐♥✈♦❧✉çã♦✱ ❝❤❛♠❛❞❛ ❞❡ ■♥✈♦❧✉çã♦ ❈❛♥ô♥✐❝❛ ✭♦✉ ❈❧áss✐❝❛✮ ❡♠ ❆♥é✐s ❞❡ ●r✉♣♦s✳ ❊ss❛ ♣❛rt✐❝✉❧❛r ✐♥✈♦❧✉çã♦ é ♦❜❥❡t♦ ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ❡♠ q✉❛❧q✉❡r ❡st✉❞♦ q✉❡ r❡❧❛❝✐♦♥❡ ❛♥é✐s ❞❡ ❣r✉♣♦s ❡ ✐♥✈♦❧✉çõ❡s✱ s❡♥❞♦ q✉❡ ❛ ♣❛rt✐r ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥❡ss❡s ❝❛s♦s✱ ❜✉s❝❛♠✲s❡ ❡st❡♥❞ê✲❧♦s ♣❛r❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ q✉❛❧q✉❡r ♦✉ ❛té ♣❛r❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ❣❡r❛❧ ❞❡ ❛♥é✐s✳ ❊①❡♠♣❧♦s ❞♦ ❡st✉❞♦ ❞❡ss❛ ✐♥✈♦❧✉çã♦ ❡ ❞❛ s✉❛ ❡①t❡♥sã♦✱ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬▲✾✾✱ ▲✵✵✱ ▲✶✵✱ ●P❙✵✾✱ ▲❙❙✵✾❪✳ ❉❡✜♥✐çã♦ ✶✳✶✾✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❡ ∗ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ R✳ ❯♠ ❡❧❡♠❡♥t♦ r ∈ R é ❝❤❛✲ ∗ ∗

  = r = ♠❛❞♦ ❞❡ s✐♠étr✐❝♦ ♦✉ ❛♥t✐ss✐♠étr✐❝♦✱ s❡ r ♦✉ r −r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡♥♦t❛♠♦s ♣♦r R ❡ R ∗ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s s✐♠étr✐❝♦s ❡ ❛♥t✐ss✐♠étr✐❝♦s ❞❡ R✱ ♦✉ s❡❥❛✱ ∗ − ∗

  R = = r = = {r ∈ R; r } ❡ R ∗ {r ∈ R; r −r} .

  • + ◆❛ ❧✐t❡r❛t✉r❛ é ❝♦♠✉♠ ❡♥❝♦♥tr❛r ❛ ♥♦t❛çã♦ R ✱ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s s✐♠étr✐✲

  ❝♦s✱ ❡ R ✱ ♣❛r❛ ♦s ❛♥t✐ss✐♠étr✐❝♦s✳ ❯t✐❧✐③❛r❡♠♦s ❛s ❞✉❛s ♥♦t❛çõ❡s ❞❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ q✉❡✱ ❡♠ ❛❧❣✉♥s ♠♦♠❡♥t♦s✱ s❡rá ✐♥t❡r❡ss❛♥t❡ ✐❞❡♥t✐✜❝❛r ❛ ✐♥✈♦❧✉çã♦ ❡♠ q✉❡stã♦ ❡✱ ♣♦rt❛♥t♦✱ ❛

  ✶✷ ♣r✐♠❡✐r❛ s❡ t♦r♥❛ ♠❛✐s ❛❞❡q✉❛❞❛✱ ❡♠ ❝♦♥tr❛♣❛rt✐❞❛✱ ❛ s❡❣✉♥❞❛ é ♠❛✐s ❡❝♦♥ô♠✐❝❛✳ ❯t✐❧✐✲ −1 ∗ −∗

  ) = x , ③❛r❡♠♦s t❛♠❜é♠ ❛s ♥♦t❛çõ❡s (x ∀x ∈ G ❡ G ❛♦ ❝♦♥❥✉♥t♦ ❞♦s s✐♠étr✐❝♦s ❞♦ ❣r✉♣♦ G✱ ♦✉ s❡❥❛✱

  G = ∗ ∗ = (xx ) = {x ∈ G : x = x } ∗ −1 ∗ ❖❜s❡r✈❛çã♦ ✶✳✷✵✳ ◆♦t❡ q✉❡ 1 ∈ G ❡ ♣♦rt❛♥t♦✱ s❡ x ∈ G ✱ ❡♥tã♦ 1 = 1 −∗ ∗ −∗ −∗ −1 −1 x x = x x = x ∗ ∗ / ∗ ∗ ∗ ✱ ❛ss✐♠✱ x ✱ ♦✉ s❡❥❛ x ∈ G ✳ ❆♥❛❧♦❣❛♠❡♥t❡ ♠♦str❛♠♦s q✉❡ −1 −1 s❡ x /∈ G ✱ ❡♥tã♦ x ∈ G ✱ ♦✉ s❡❥❛✱ x ∈ G s❡✱ ❡ s♦♠❡♥t❡ s❡ x ∈ G ✳ ❆❞❡♠❛✐s −∗ ∗ −1 x = (x )

  ✳ ❯t✐❧✐③❛r❡♠♦s ♦s ❢❛t♦ ❛❝✐♠❛ ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦ s❡♠ ❢❛③❡r r❡❢❡rê♥❝✐❛ ❛ ❡ss❛ ♦❜s❡r✈❛✲

  çã♦✳ X ❙❡❥❛ RG ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦ ❝♦♠ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ✐♥❞✉③✐❞❛ ♣♦r G✱ ❛ss✐♠✱ ❞❛❞♦ α = α x x x ∈G ∈ RG✱ t❡♠♦s ∗ ∗ X

  α = α x x , x ∈G + ♣♦rt❛♥t♦✱ s❡ α ∈ RG ✱ ❡♥tã♦ X X x ∈G x ∈G α x x = α x x . +

  ❉❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ RG s❡r ❧✐✈r❡♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠♦ R✲♠ó❞✉❧♦ ♣♦r G✱ ♣❛rt✐❝✐♦♥❛♥❞♦ G = G ) ∗ ∗ x = α x

  ∪ (G\G ✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ s❡ α ∈ RG ✱ ❡♥tã♦ α s❡ x /∈ G ❀ ❛❞❡♠❛✐s✱ ♥❡♥❤✉♠❛ ❝♦♥❞✐çã♦ é ❡①✐❣✐❞❛ ♣❛r❛ ♦s ❝♦❡✜❝✐❡♥t❡s ❛ss♦❝✐❛❞♦s ❛ ❡❧❡♠❡♥t♦s ❞❡ G ✱ ❧♦❣♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ RG é ❣❡r❛❞♦ ❝♦♠♦ ✉♠ R✲♠ó❞✉❧♦ ♣♦r +

  G : x ∗ ∪ {x + x ∈ G\G ∗ } . ❉❡ ✉♠❛ ❢♦r♠❛ ❛♥á❧♦❣❛✱ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡ RG é ❣❡r❛❞♦ ♣♦r ∗ ∗ , r 2 : x 2 {rx : x ∈ G ∈ R } ∪ {x − x ∈ G\G } ,

  = ♦♥❞❡ R {r ∈ R : 2r = 0}✳

  ❖ ✐♥t❡r❡ss❡ ❡♠ ❡st✉❞❛r ❡ss❡s ❝♦♥❥✉♥t♦s ❢♦✐ ✐♠♣✉❧s✐♦♥❛❞♦ ❛♣ós ❆♠✐ts✉r ♣r♦✈❛r✱ ❡♠ + ❬❆✻✾❪✱ q✉❡ s❡ R ♦✉ R s❛t✐s❢❛③ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧✱ ❡♥tã♦ R t❛♠❜é♠ s❛t✐s❢❛③ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡✳ ◆❛t✉r❛❧♠❡♥t❡✱ ❡♠ ❣❡r❛❧✱ ❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❞❡♠ ♥ã♦ s❡r ♣r❡s❡r✈❛❞❛s ❞♦s s✐♠étr✐❝♦s ♣❛r❛ t♦❞♦ ♦ ❛♥❡❧ R✱ s❡♥❞♦ ❛ss✐♠✱ ✉♠ ♣r♦❜❧❡♠❛ q✉❡ s✉r❣❡ ✐♠❡❞✐❛t❛♠❡♥t❡ + ❞❡ss❡ r❡s✉❧t❛❞♦ é ❝♦♥❤❡❝❡r q✉❛✐s ✐❞❡♥t✐❞❛❞❡s ✈❡r✐✜❝❛❞❛s ❡♠ R ♦✉ R t❛♠❜é♠ ♦ s❡rã♦ + ❡♠ R✱ ♥❡ss❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❢♦✐ ❧❡✈❛♥t❛❞❛ ❞❡ R ♦✉ R ♣❛r❛ R✳

  ❉❡♥♦t❛♥❞♦ ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ ❡♠ ✉♠ ❛♥❡❧ R ♣♦r [r, s] = rs−sr, ∀r, s ∈ R✱ ❛ ♣❛rt✐r ❞❛ ✐t❡r❛❞❛ ❞❡ss❡s ❝♦❧❝❤❡t❡s✱

  i ✶✸

  ∀r ∈ R✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛s s❡❣✉✐♥t❡s ✐❞❡♥t✐❞❛❞❡s 1 [r , . . . , r n ] = 0

  ❡ [r, s, s, . . . , s ] = 0, | {z } n −

  

✈❡③❡s

  ❝❤❛♠❛❞❛s ❞❡ ▲✐❡ ♥✐❧♣♦tê♥❝✐❛ ❞❡ í♥❞✐❝❡ n ❡ ▲✐❡ n✲❊♥❣❡❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ◆♦ ❝❛s♦ ❞❡ K s❡r ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡ 2✱ ●✐❛♠❜r✉♥♦✱ ❙❡❤❣❛❧

  ❡ ▲❡❡✱ ♣r♦✈❛r❛♠✱ ❡♠ ❬●❙✾✸✱ ▲✵✵❪✱ q✉❡ s❡ G é ✉♠ ❣r✉♣♦ q✉❡ ♥ã♦ ♣♦ss✉✐ 2✲❡❧❡♠❡♥t♦s ❡ ∗ é ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛ ❡♠ G✱ ❡♥tã♦ ❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ ▲✐❡ ♥✐❧♣♦tê♥❝✐❛ ❡ ▲✐❡ n✲❊♥❣❡❧ ♣♦❞❡♠ + s❡r ❧❡✈❛♥t❛❞❛s ❞❡ KG ❡ KG ♣❛r❛ t♦❞♦ ♦ KG✳ P♦st❡r✐♦r♠❡♥t❡ ●✐❛♠❜r✉♥♦✱ P♦❧❝✐♥♦ ❡ ❙❡❤❣❛❧✱ ♠♦str❛r❛♠✱ ❡♠ ❬●P❙✵✾❪✱ q✉❡ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s s✐♠étr✐❝♦s ❛ ❝♦♥❞✐çã♦ ❞❡ G

  ♥ã♦ ♣♦ss✉✐r 2✲❡❧❡♠❡♥t♦s ❝♦♥t✐♥✉❛ s❡♥❞♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❣❛r❛♥t✐r q✉❡ ❡ss❛s ✐❞❡♥t✐❞❛❞❡s s❡❥❛♠ ❧❡✈❛♥t❛❞❛s ♥♦ ❝❛s♦ ❞❛ ✐♥✈♦❧✉çã♦ ∗ s❡r ✉♠❛ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ q✉❛❧q✉❡r ❡♠ KG❀ ❛❧é♠ ❞✐ss♦✱ ▲❡❡✱ ❡♠ ❬▲✾✾❪✱ ❡st❡♥❞❡✉ ✉♠ ❞♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ❡♠ ❬●❙✾✸❪✱ ❣❛r❛♥t✐♥❞♦ q✉❡ ♥♦ ❝❛s♦ ❞❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✱ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❛✐♥❞❛ é ✈❡r✐✜❝❛❞♦ ❝❛s♦ ♦ ❣r✉♣♦ ❞♦s 8 q✉❛tér♥✐♦s Q ♥ã♦ ❡st❡❥❛ ❝♦♥t✐❞♦ ❡♠ G✳ +

  P♦r ♦✉tr♦ ❧❛❞♦✱ q✉❛♥❞♦ ♦s ❝♦♥❥✉♥t♦s RG ❡ RG ✈❡r✐✜❝❛♠ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ❡ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ♥ã♦ ♦❝♦rr❡✱ é ♣♦ssí✈❡❧ ❛✐♥❞❛ ❡①tr❛✐r ✐♥❢♦r♠❛çõ❡s ❛❝❡r❝❛ ❞❛ ❡str✉t✉r❛ ❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG✱ ❜✉s❝❛♥❞♦ ❝❛r❛❝t❡r✐③❛r ♦ ❣r✉♣♦ G q✉❡ ♦ ❝♦♠♣õ❡✱ ❛ss✐♠ ❝♦♠♦ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❛♥❡❧ R✳ ◆❡ss❡ s❡♥t✐❞♦ ♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ❡♠ ❬▲✾✾✱ ▲✵✵✱ ▲❙❙✵✾❪ ❝♦♠♣❧❡♠❡♥t❛♠ ♦ ❡st✉❞♦ ❝♦♠❡♥t❛❞♦ ♥♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✳ ❉❡ ✉♠❛ ❢♦r♠❛ r❡s✉♠✐❞❛✱ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛ ❧♦❝❛❧✐③❛ ♦s r❡s✉❧t❛❞♦s q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❛❜♦r❞❛r✿

  ▲✐❡ ♥✐❧♣♦tê♥❝✐❛ ▲✐❡ n✲❊♥❣❡❧ RG ∗ ❬●❙✾✸✱ ▲✾✾✱ ●P❙✵✾✱ ▲❙❙✵✾❪ ❬▲✵✵✱ ●P❙✵✾❪

  RG ∗ ❬●❙✾✸❪ ❬▲✵✵❪ ❚❛❜❡❧❛ ✶✳✶✿ Pr♦♣r✐❡❞❛❞❡s ❞❡ ▲✐❡ ♣❛r❛ ✐♥✈♦❧✉çõ❡s ✐♥❞✉③✐❞❛s

  ❊①✐st❡♠ ♦✉tr❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ ✐♥t❡r❡ss❡ ♥❛ t❡♦r✐❛✱ ❞❡♥tr❡ ❡❧❛s✱ ❞❡st❛❝❛r❡♠♦s ❛q✉❡✲ + ❧❛s q✉❡ ❡stã♦ ❞✐r❡t❛♠❡♥t❡ ❧✐❣❛❞❛s à ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♠✉♥✐r ♦s ❝♦♥❥✉♥t♦s R ❡ R ❞❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❛♥❡❧✳ ❉❡✜♥✐çã♦ ✶✳✷✶✳ ❉✐③❡♠♦s q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A ⊂ R é ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡ ab = −ba✱ ∀a, b ∈ A✱ ♦✉ s❡❥❛✱ ♦ ♦♣❡r❛❞♦r ❞❡ ❏♦r❞❛♥ a ◦ b = ab + ba é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ❡♠ A✳ +

  ❖ ✐♠♣❡❞✐♠❡♥t♦ ♣❛r❛ q✉❡ ♦s ❝♦♥❥✉♥t♦s R ❡ R ❤❡r❞❡♠ ❛ ❡str✉t✉r❛ ❞❡ ❛♥❡❧ ❞❡ R✱ s❡ ❡♥❝♦♥tr❛ ♥♦ ❢❛t♦ ❞❡ q✉❡ ❡st❡s ♣♦❞❡♠ ♥ã♦ s❡r ❢❡❝❤❛❞♦s ♣❛r❛ ♦ ♣r♦❞✉t♦✱ ❡ é ❥✉st❛♠❡♥t❡ s♦❜r❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ q✉❡ r❡♣♦✉s❛♠ ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ❡ss❡s

  ✶✹ Pr♦♣♦s✐çã♦ ✶✳✷✷✳ ❙❡ R ✉♠ ❛♥❡❧ ❝♦♠ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗✱ ❡♥tã♦ sã♦ ✈❡r❞❛❞❡✐r❛s✿ + +

  ✭❛✮ R é ✉♠ s✉❜❛♥❡❧ ❞❡ R s❡✱ ❡ s♦♠❡♥t❡ s❡✱ R é ❝♦♠✉t❛t✐✈♦✳ ✭❜✮ R é ✉♠ s✉❜❛♥❡❧ ❞❡ R s❡✱ ❡ s♦♠❡♥t❡ s❡✱ R é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ∗ ∗ ∗ + +

  = a = a ❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❙❡❥❛♠ a, b ∈ R ✳ ❊♥tã♦ (a − b) ∗ ∗ ∗ − b − b✳ ▲♦❣♦ a − b ∈ R ✳ +

  = b a = ba ❆❣♦r❛✱ (ab) ✳ ❆ss✐♠✱ ab ∈ R s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ab = ba❀ ✐st♦ é✱ s❡✱ ❡ s♦♠❡♥t❡ + s❡✱ R é ❝♦♠✉t❛t✐✈♦✳ ❖ ✐t❡♠ ✭❜✮ s❡❣✉❡ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✳

  ❊ss❛ ♣r♦♣♦s✐çã♦ ♠♦t✐✈❛ ❛ ❜✉s❝❛ ♣♦r r❡s✉❧t❛❞♦s q✉❡ r❡❧❛❝✐♦♥❡♠ ❡st❛s ✐❞❡♥t✐❞❛❞❡s s♦❜r❡ ❡ss❡s s✉❜❝♦♥❥✉♥t♦s✱ q✉❡ é ❡①❛t❛♠❡♥t❡ ♦ ♣r♦♣ós✐t♦ ❞❡ss❛ t❡s❡ ♣❛r❛ ❝❡rt❛s ✐♥✈♦❧✉çõ❡s q✉❡ ✐r❡♠♦s ❡♠ ❜r❡✈❡ ❛❜♦r❞❛r✳

  P♦❞❡♠♦s ❡♥❝♦♥tr❛r ♥❛ ❝r❡s❝❡♥t❡ ❧✐t❡r❛t✉r❛ ❛❝❡r❝❛ ❞❡ss❡ t❡♠❛✱ r❡s✉❧t❛❞♦s ❣❛r❛♥✲ t✐♥❞♦✱ ♣❛r❛ ❛ ✐♥✈♦❧✉çã♦ ∗ ✐♥❞✉③✐❞❛ ❡♠ RG✱ ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦✲ ♠✉t❛t✐✈✐❞❛❞❡ ❞❡ RG ❡ RG ∗ s❡❥❛♠ ✈❡r✐✜❝❛❞❛s✳ ❚❛✐s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s✱ ♥♦ ❝❛s♦ charR 6= 2✱ ❝♦♥❢♦r♠❡ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿

  ❈♦♠✉t❛t✐✈✐❞❛❞❡ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ RG

  ❬❏❘✵✻❪ ❬●P✶✸❛❪ RG ∗ ❬❇❏P❘✵✾❪

  ❚❛❜❡❧❛ ✶✳✷✿ ❈♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ♣❛r❛ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s + ❖❜s❡r✈❡ q✉❡ s❡ charR = 2✱ ❡♥tã♦ ♦s ❝♦♥❥✉♥t♦s RG ❡ RG ❝♦✐♥❝✐❞❡♠✱ ❡ ♦ ♠❡s♠♦

  ♦❝♦rr❡ ♣❛r❛ ❡ss❛s ✐❞❡♥t✐❞❛❞❡s✳ ❊ss❡ ❝❛s♦ ❢♦✐ ❡st✉❞❛❞♦ ❡♠ [❏❘✵✻]✳

  ✶✳✸ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s

  ❖❜s❡r✈❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❝✐t❛❞♦s ❛❝✐♠❛ ❛❝❡r❝❛ ❞❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ ▲✐❡ ♥✐❧♣♦tê♥❝✐❛✱ + ▲✐❡ n✲❊♥❣❡❧✱ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ✈❡r✐✜❝❛❞❛s ♥♦s ❝♦♥❥✉♥t♦s RG ❡ RG ✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✱ ♥✉♠ ❝❡rt♦ s❡♥t✐❞♦✱ ♣♦✉❝♦ t❡♠✲s❡ ❛ ❛❝r❡s❝❡♥t❛r ♥❡ss❡ ❡st✉❞♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ ♣r♦ss❡❣✉✐r♠♦s ❞❡✈❡♠♦s ❜✉s❝❛r ♥♦✈❛s ❢♦r♠❛s ❞❡ ❝♦♥str✉✐r ✐♥✈♦❧✉çõ❡s ♥♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG ❛ ♣❛rt✐r ❞❡ ✐♥✈♦❧✉çõ❡s ❡♠ G✳ ◆❛ ❜✉s❝❛ ❞❡ss❛s ✐♥✈♦❧✉çõ❡s✱ s✉r❣❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s✱ ✐♥tr♦❞✉③✐❞♦s ♣♦r ◆♦✈✐❦♦✈ ♥♦ ❝♦♥t❡①t♦ ❞❛ K✲❚❡♦r✐❛✳ ❉❡✜♥✐çã♦ ✶✳✷✸✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❡ G ✉♠ ❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ∗✳ ❙❡❥❛ σ : G → {±1} ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✱ ♦ q✉❛❧ ❝❤❛♠❛♠♦s ❞❡ ♦r✐❡♥t❛çã♦ ❞❡ G✱ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦ ∗✱ ♥♦ s❡♥t✐❞♦ ❞❡ xx ∈ N := kerσ✳ ❊♥tã♦✱ ❛ ❛♣❧✐❝❛çã♦ σ∗ : RG → RG ❞❛❞❛ ♣♦r X ! σ ∗ X x ∈G x ∈G α x x = σ(x)α x x ,

  σ σ ∗ ∗ ✶✺ ) = x

  ◆♦t❡ q✉❡ s❡ σ∗ ❞❡✜♥✐❞❛ ❛❝✐♠❛ é ✉♠❛ ✐♥✈♦❧✉çã♦✱ ❞❡✈❡♠♦s t❡r (x ✱ ♦✉ s❡❥❛✱ σ σ σ ∗ ∗ ∗ ∗ ∗ (x ) = (σ(x)x ) = σ(xx )x = x,

  ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡✱ ❞❡ ❢❛t♦✱ ❛ ❝♦♥❞✐çã♦ xx ∈ N é ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ ❜♦❛ ❞❡✜♥✐çã♦ ❞❡ss❛ ✐♥✈♦❧✉çã♦✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ❡ss❛ ♠❡s♠❛ ❝♦♥❞✐çã♦ t❛♠❜é♠ é s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ σ∗ s❡❥❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ RG✳ ❖❜s❡r✈❛çã♦ ✶✳✷✹✳ ❙❡ σ ❢♦r tr✐✈✐❛❧✱ ♦✉ s❡❥❛✱ σ ≡ 1✱ ❡♥tã♦ ❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ σ∗ ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ ∗✳ ❉❡ss❛ ❢♦r♠❛✱ t♦❞❛s ❛s ♦r✐❡♥t❛çõ❡s s❡rã♦ ❛ss✉♠✐❞❛s ♥ã♦ tr✐✈✐❛✐s✳ ◆♦t❡ s❡ σ∗ é ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❝♦♠ σ ♥ã♦ tr✐✈✐❛❧✱ ❡♥tã♦ charR 6= 2✳ ❉❡st❛ ❢♦r♠❛ ❛ss✉♠✐r❡♠♦s t❛♠❜é♠ ❞✉r❛♥t❡ t♦❞♦ ♦ tr❛❜❛❧❤♦ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ charR 6= 2✳

  ❖s ♣ró①✐♠♦s ❡①❡♠♣❧♦s ♠♦str❛♠ q✉❡ ❡①✐st❡ ✉♠❛ ❣r❛♥❞❡ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥✈♦❧✉çõ❡s ❝♦♠♣❛tí✈❡✐s ❝♦♠ ♦r✐❡♥t❛çõ❡s✳ ❊①❡♠♣❧♦ ✶✳✷✺✳ ❚♦♠❛♥❞♦ ♦ ❣r✉♣♦ GL(n, K) ❝♦♠ ❛ ✐♥✈♦❧✉çã♦ ∗ ❞❛❞❛ ♣❡❧❛ tr❛♥s♣♦s✐çã♦ detA ❞❡ ♠❛tr✐③❡s✱ t♦♠❛♥❞♦ σ(A) = ✱ A ∈ GL(n, K)✱ ❢❛❝✐❧♠❡♥t❡ ❝♦♥❝❧✉í♠♦s q✉❡ σ ❡ ∗ sã♦ |detA| ❝♦♠♣❛tí✈❡✐s ❡ ♣♦rt❛♥t♦ σ∗ ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ KGL(n, K)✳ ❉❛ ♠❡s♠❛ ♠❛♥❡✐r❛✱ é ❢á❝✐❧ ✈❡r q✉❡ σ∗ ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ KH✱ ❝♦♠ H ❞❛❞♦ ♥♦ ❊①❡♠♣❧♦ ✶✳✻✳ ❊①❡♠♣❧♦ ✶✳✷✻✳ ❈♦♠♦ ✉♠❛ ♦r✐❡♥t❛çã♦ σ é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ G ❡♠ {±1}✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ❡①✐st❡ ✉♠❛ r❡❧❛çã♦ ❜✐✉♥í✈♦❝❛ ❡♥tr❡ ❛s ♦r✐❡♥t❛çõ❡s ❞❡ ✉♠ ❣r✉♣♦ G ❡ ♦s s✉❜✲ ❣r✉♣♦s N ≤ G t❛✐s q✉❡ (G : N) = 2✳ P❛r❛ ✐st♦✱ ❜❛st❛ ✈❡r✐✜❝❛r q✉❡ t♦♠❛♥❞♦ kerσ = N✱ ❡♥tã♦ (G : N) = 2❀ r❡❝✐♣r♦❝❛♠❡♥t❡✱ ♣❛r❛ ✉♠ ❞❛❞♦ N ≤ G t❛❧ q✉❡ (G : N) = 2✱ ❞❡✜♥✐♥❞♦ σ : G

  → G ♣♦r ( −1 , s❡ x / ∈ N

  σ(x) = 1 , s❡ x ∈ N, t❡♠♦s ❛ r❡❧❛çã♦ ❞❡s❡❥❛❞❛✳ ◆♦t❡ q✉❡ ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ❣❛r❛♥t❡ q✉❡ ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛ ❡♠

  ✉♠ ❣r✉♣♦ G é ❝♦♠♣❛tí✈❡❧ ❝♦♠ q✉❛❧q✉❡r ♦r✐❡♥t❛çã♦ ❞❡st❡✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r ♠♦str❛ q✉❡ ❞❛❞♦s ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ q✉❛❧q✉❡r

  ❡ ✉♠❛ ♦r✐❡♥t❛çã♦ ❡♠ ✉♠ ❣r✉♣♦ G✱ ♥❡♠ s❡♠♣r❡ ❡st❛s s❡rã♦ ❝♦♠♣❛tí✈❡✐s✳ 4 ❊①❡♠♣❧♦ ✶✳✷✼✳ ❉❛❞♦ ♦ ❣r✉♣♦ D ❝♦♠ ✐♥✈♦❧✉çã♦ ∗ ❞❡✜♥✐❞❛ ♥♦ ❊①❡♠♣❧♦ ✶✳✼✱ t♦♠❛♥❞♦ ❛ ♦r✐❡♥t❛çã♦ σ(x) = −1✱ σ(y) = 1✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ∗ ♥ã♦ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ σ ❡ 4

  ♣♦rt❛♥❞♦ ❛ ❛♣❧✐❝❛çã♦ σ∗ ❞❡✜♥✐❞❛ ❛❝✐♠❛ ♥ã♦ é ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ RD ✳ ❉❛❞♦ ❡ss❡ ♥♦✈♦ t✐♣♦ ❞❡ ✐♥✈♦❧✉çã♦✱ s✉r❣❡♠ ♥❛t✉r❛❧♠❡♥t❡ ❛s ♠❡s♠❛s q✉❡stõ❡s ❥á

  ❡st✉❞❛❞❛s ♥♦ ❝❛s♦ ♥ã♦ ♦r✐❡♥t❛❞♦✳ ❆❧é♠ ❞✐ss♦✱ t♦❞♦ ♦ ❡st✉❞♦ ❢❡✐t♦ ❛♥t❡r✐♦r♠❡♥t❡ ♠♦str❛✲s❡ ❜❛st❛♥t❡ út✐❧ ♥❛ r❡s♦❧✉çã♦ ❞❡ss❡s ♥♦✈♦s ♣r♦❜❧❡♠❛s✱ ✈✐st♦ q✉❡✱ ❞❛❞❛ ❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ σ RN : RN

  ∗✱ ❢❛❝✐❧♠❡♥t❡ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ σ ∗ | → RN é ♣r❡❝✐s❛♠❡♥t❡ ❛ ✐♥✈♦❧✉çã♦

  σ ∗ σ ✶✻ ✐♥❞✉③✐❞❛ ♣♦r ∗ ❡♠ RN❀ ❞❡ss❛ ❢♦r♠❛✱ s❡ RG ✭♦✉ RG ∗ ✮ ✈❡r✐✜❝❛ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐✲ σ = RN ∗ ∗ ♥♦♠✐❛❧✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ RN ✭♦✉ RN ∗ ✮ ✐rá s❛t✐s❢❛③❡r ❛ ♠❡s♠❛ ✐❞❡♥t✐❞❛❞❡✱ ❧♦❣♦✱ s❡ ❢♦r ❝♦♥❤❡❝✐❞❛ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s s✐♠étr✐❝♦s ✭♦✉ ❛♥t✐ss✐♠étr✐✲ ❝♦s✮ s❛t✐❢❛③ ❡ss❛ ✐❞❡♥t✐❞❛❞❡✱ ❡♥tã♦✱ ❝♦♥❤❡❝❡r❡♠♦s ♦ s✉❜❣r✉♣♦ N ≤ G ❡✱ ♣♦rt❛♥t♦✱ ❣r❛♥❞❡ ♣❛rt❡ ❞♦ ❣r✉♣♦ G é ❝♦♥❤❡❝✐❞❛✱ ✈✐st♦ q✉❡ (G : N) = 2✳

  ❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ✜③❡♠♦s ♣❛r❛ ♦ ❝❛s♦ ♥ã♦ ♦r✐❡♥t❛❞♦✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r ✉♠ + ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s ♣❛r❛ RG ❡ RG ✳ ■r❡♠♦s ❞❡s❝r❡✈❡r ❡ss❡s ❝♦♥❥✉♥t♦s ♥❛ ♣ró①✐♠❛ s❡çã♦ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♠❛✐s ❣❡r❛❧✳

  ❋✐♥❛❧♠❡♥t❡✱ ✉t✐❧✐③❛♥❞♦ ♦s r❡s✉❧t❛❞♦s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ♦ s✉❜❣r✉♣♦ N✱ ✈✐❞❡ ❛ ❚❛✲ σ ❜❡❧❛ ✶✳✷✱ ❡ ❛ ❞❡s❝r✐çã♦ ❞♦s ❣❡r❛❞♦r❡s ❞❡ RG ❡ RG σ ∗ ✱ ❢♦✐ ♣♦ssí✈❡❧ q✉❛s❡ q✉❡ ❝♦♠♣❧❡t❛✲ ♠❡♥t❡ ❡s❣♦t❛r ♦ ❡st✉❞♦ ❞❡ss❡ t✐♣♦ ❞❡ ✐♥✈♦❧✉çã♦ ❡♠ r❡❧❛çã♦ ❛s ✐❞❡♥t✐❞❛❞❡s ✉s✉❛✐s✱ r❡st❛♥❞♦ ❛♣❡♥❛s ❞❡s❝r❡✈❡r q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❛♥t✐ss✐♠étr✐❝♦s ❝♦♠✉t❛♠ s❡♥❞♦ ∗ ✉♠❛ ✐♥✈♦❧✉çã♦ q✉❛❧q✉❡r✱ ✈✐st♦ q✉❡ ❡ss❡ ❝❛s♦ ❢♦✐ ❛❜♦r❞❛❞♦ ❛♣❡♥❛s ♣❛r❛ ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✳ ❊ss❡ ú❧✲ t✐♠♦ ♣r♦❜❧❡♠❛ ❡♠ ❛❜❡rt♦ é ♦ tó♣✐❝♦ ❞♦ ❈❛♣ít✉❧♦ ✹✱ ♦♥❞❡ ✐r❡♠♦s ❛♣r❡s❡♥t❛r ✉♠❛ r❡s♣♦st❛ ♣❛r❝✐❛❧ ♣❛r❛ ❡ss❡ ❝♦♥❞✐çã♦✳ ❆♦s ❞❡♠❛✐s ❝❛s♦s✱ ❡ ❛ r❡s♣♦st❛ ❞❡ q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❛♥t✐ss✐♠étr✐❝♦s ❝♦♠✉t❛♠ s❡♥❞♦ ∗ ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✱ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠

  ❈♦♠✉t❛t✐✈✐❞❛❞❡ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ Pr♦♣r✐❡❞❛❞❡s ❞❡ ▲✐❡ RG σ

  ❬❇P✵✻✱ ●P✶✸❜❪ ❬●P✶✹❪ ❬❈P✶✷✱ ❱✶✸❪

  RG σ ❬❇❏❘✵✼❪ ❬●P✶✸❜❪ ❚❛❜❡❧❛ ✶✳✸✿ ■❞❡♥t✐❞❛❞❡s ♣❛r❛ ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s

  ❯♠ ❢❛t♦ ✐♥t❡r❡ss❛♥t❡ q✉❡ ♦❝♦rr❡ ❛♦ ❝♦♠♣❛r❛r♠♦s ♦s r❡s✉❧t❛❞♦s ❛❝❡r❝❛ ❞❛s ✐♥✈♦❧✉✲ çõ❡s ✐♥❞✉③✐❞❛s ❡ ♦r✐❡♥t❛❞❛s✱ é q✉❡✱ ❛ ❡str✉t✉r❛ ❞♦s ❣r✉♣♦s ❡ ❞❛ ✐♥✈♦❧✉çã♦✱ ♦✉ ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❛ ✐❞❡♥t✐❞❛❞❡✱ sã♦✱ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❝❛s♦s✱ ❛s ♠❡s♠❛s ❞♦ ❝❛s♦ ♦r✐❡♥t❛❞♦ ❡ ♦ ✐♥❞✉③✐❞♦✳

  ❯♠ ❡①❡♠♣❧♦ ❞❡ss❡ ❢❛t♦ ♣♦❞❡ s❡r ✈✐st♦ ♥♦ ❡st✉❞♦ ❞❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❡❧❡✲ ♠❡♥t♦s s✐♠étr✐❝♦s ❡ ❛♥t✐ss✐♠étr✐❝♦s✱ ♦♥❞❡ t❛♥t♦ ♦s ❝❛s♦s ♦r✐❡♥t❛❞♦s ❡ ♥ã♦ ♦r✐❡♥t❛❞♦s ❛♣r❡✲ s❡♥t❛♠ ❝♦♠♦ s♦❧✉çõ❡s ❣r✉♣♦s q✉❡ ♣♦ss✉❡♠ ♥♦ ♠á①✐♠♦ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧✱ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❛ ✐♥✈♦❧✉çã♦ ♥♦s ❡❧❡♠❡♥t♦s ♥ã♦ s✐♠étr✐❝♦s ❡♠ G é ❞❛❞❛ ♣r❡❝✐s❛♠❡♥t❡ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ s ✜①♦ ❞❡ G✱ q✉❡✱ ♥♦ ❝❛s♦ ♥ã♦ ❛❜❡❧✐❛♥♦✱ é ♦ ♣ró♣r✐♦ ❣❡r❛❞♦r ❞❡ G ✳ ◆♦ ❝❛s♦ ❞❛ ▲✐❡ ♥✐❧♣♦tê♥❝✐❛ ♦✉ ▲✐❡ n✲❊♥❣❡❧✱ ❛ ❛✉sê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ♦r❞❡♠ ❞♦✐s é ❢❛t♦r ❝♦♠✉♠ ♣❛r❛ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❡ss❛s ✐❞❡♥t✐❞❛❞❡s✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞❛ ✐♥✈♦❧✉çã♦ s❡r ♦r✐❡♥t❛❞❛ ♦✉ ♥ã♦✳

  ✶✳✹ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s

  ◆♦ ✐♥t✉✐t♦ ❞❡ ❡st❡♥❞❡r ❛✐♥❞❛ ♠❛✐s ❛ ❝♦♥str✉çã♦ ❞❡ ✐♥✈♦❧✉çõ❡s ❡♠ ❛♥é✐s ❞❡ ❣r✉♣♦s

  ✶✼ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ {±1} ❞❛ ♦r✐❡♥t❛çã♦ ✉s✉❛❧ σ ♣❡❧♦ ❣r✉♣♦ ❞❛s ✉♥✐❞❛❞❡s ❞♦ ❛♥❡❧ R✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ ✉♠❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛ ❡stá ❢♦rt❡♠❡♥t❡ ❧✐❣❛❞❛ ❛♦ ❛♥❡❧ R ❜❛s❡ ❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG s♦❜r❡ ♦ q✉❛❧ ♣r❡t❡♥❞❡♠♦s ❝♦♥str✉✐r ❡ss❛s ♥♦✈❛s ✐♥✈♦❧✉çõ❡s✳ ❚♦♠❛♥❞♦ ❡ss❛ ♥♦✈❛ ♦r✐❡♥t❛çã♦✱ ♣♦❞❡♠♦s✱ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ às ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s✱ ❞❡✜♥✐r ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ RG✳

  ❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ✈❡r✐✜❝❛♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ ❛ ❝♦♥❞✐çã♦ xx ∈ N = kerσ é ♥❡✲ ❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ ❛ ❛♣❧✐❝❛çã♦ ❛ s❡❣✉✐r s❡❥❛ ✉♠❛ ✐♥✈♦❧✉çã♦✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ t❡♠♦s✿ ❉❡✜♥✐çã♦ ✶✳✷✽✳ ❙❡❥❛ RG ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦ ❞❡ R s♦❜r❡ ✉♠ ❣r✉♣♦ G ❝♦♠ ✐♥✈♦❧✉çã♦ ∗✳ ❯♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s σ : G → U(R) é ❝❤❛♠❛❞♦ ❞❡ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛ ❞❡ G✳ ❙❡ ∗ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛ σ ♥♦ s❡♥t✐❞♦ ❞❡ xx ∈ N := kerσ✱ ❡♥tã♦ ❛ ❛♣❧✐❝❛çã♦ σ∗ : RG → RG ❞❛❞❛ X ! ∗ σ X x x ∈G ∈G α x x = σ(x)α x x

  ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♥♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG✱ ❝❤❛♠❛❞❛ ❞❡ ■♥✈♦❧✉çã♦ ❖r✐❡♥t❛❞❛ ●❡♥❡r❛❧✐✲ ③❛❞❛✳ ❖❜s❡r✈❛çã♦ ✶✳✷✾✳ ❆✜♠ ❞❡ t♦r♥❛r ❛ ❡s❝r✐t❛ ♠❛✐s ❧✐♠♣❛✱ q✉❛♥❞♦ ♥ã♦ ❤♦✉✈❡r r✐s❝♦ ❞❡ ❝♦♥❢✉sã♦✱ ✐r❡♠♦s ♥♦s r❡❢❡r✐r ❛♣❡♥❛s ♣♦r ♦r✐❡♥t❛çã♦ ❡ ✐♥✈♦❧✉çã♦ à ❡ss❛s ♥♦✈❛s ❛♣❧✐❝❛çõ❡s✳

  ◆♦ ❝❛s♦ ❞❡ ✉♠❛ ♦r✐❡♥t❛çã♦ ✉s✉❛❧✱ ✈✐♠♦s q✉❡✱ ♣❛r❛ q✉❡ ❡st❛ ♥ã♦ s❡❥❛ tr✐✈✐❛❧✱ é ♥❡✲ ❝❡ssár✐♦ q✉❡ charR 6= 2✱ ❡♥tr❡t❛♥t♦ ❡ss❛ ❝♦♥❞✐çã♦ ♥ã♦ é ❡①✐❣✐❞❛ ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ✐r❡♠♦s ❡st✉❞❛r ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✈✐✲ + ❞❛❞❡ ❞❡ RG ❡ RG ✱ q✉❡r❡♠♦s ❛ ♣r✐♥❝í♣✐♦ q✉❡ ❡st❡s ❝♦♥❥✉♥t♦s ❡ ❡st❛s ✐❞❡♥t✐❞❛❞❡s s❡❥❛♠ ❞✐st✐♥t❛s✱ ♣♦rt❛♥t♦✱ ❡♠ t♦❞♦ ♦ tr❛❜❛❧❤♦ R s❡rá ❛❞♠✐t✐❞♦ ❝♦♠ charR 6= 2✳

  P❛r❛ ❡st✉❞❛r ❡ss❛ ♥♦✈❛ ✐♥✈♦❧✉çã♦✱ ♣♦❞❡♠♦s s❡❣✉✐r ♦ ♠❡s♠♦ ❝❛♠✐♥❤♦ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❡st✉❞♦ ❞❛s ♦r✐❡♥t❛çõ❡s ✉s✉❛✐s✳ ◆❡ss❡ s❡♥t✐❞♦✱ ✉t✐❧✐③❛r❡♠♦s ❛ ❡str✉t✉r❛ ❞♦ s✉❜❣r✉♣♦ N

  ≤ G✱ ❥á ❡st❛❜❡❧❡❝✐❞❛ ❛ ♣❛rt✐r ❞♦ ❡st✉❞♦ ❞♦s ❝❛s♦s ♥ã♦ ♦r✐❡♥t❛❞♦s✱ ❡✱ ❝❛s♦ s❡❥❛ ♥❡❝❡ssár✐♦✱ t❡♠♦s t❛♠❜é♠ à ❞✐s♣♦s✐çã♦ ❛ ❡str✉t✉r❛ ❞♦ s✉❜❣r✉♣♦ C ≤ G ❞❛❞♦ ♣♦r C =

  {x ∈ G : σ(x) = ±1} , q✉❡ ❡ss❡♥❝✐❛❧♠❡♥t❡ é ♦ s✉❜❣r✉♣♦ ❞❡ G ❞❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s t❛✐s q✉❡ ❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛✲ ❧✐③❛❞❛ s❡ ❝♦♠♣♦rt❛ ❝♦♠♦ ❛ ✉s✉❛❧✳ ➱ ❢á❝✐❧ ✈❡r q✉❡ N ≤ C ≤ G✳ ❆ ❡str✉t✉r❛ ❞♦ s✉❜❣r✉♣♦ C ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ✈✐❛ ❛ ❚❛❜❡❧❛ ✶✳✸✱ ♦♥❞❡ s❛❜❡♠♦s q✉❡ ❛✐♥❞❛ ❡①✐st❡ ✉♠❛ ❧❛❝✉♥❛ ♥♦ ❝❛s♦ ❣❡r❛❧ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❛♥t✐ss✐♠étr✐❝♦s✱ ❝❛s♦ ❡ss❡ q✉❡ ✐r❡♠♦s ❛❜♦r❞❛r ♥♦ ❈❛♣ít✉❧♦ ✹✳ +

  ■r❡♠♦s ❛❣♦r❛ ❞❡s❝r❡✈❡r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s ♣❛r❛ ♦s ❝♦♥❥✉♥t♦s RG ❡ RG r❡❧❛t✐✈♦s ❛ ❡ss❛ ♥♦✈❛ ✐♥✈♦❧✉çã♦✳

  X ✶✽ +

  β x x ❙❡❥❛ β = ✳ ❙❡ β ∈ RG ❡♥tã♦ x ∈G X X ! ∗ σ X x = σ(x)β x , x x x ∈G ∈G ∈G β x x = β x x = σ(x)β x x ,

  ♣♦rt❛♥t♦ β ∀x ∈ supp(β)✳ ❉❡ss❛ ❢♦r♠❛ t❡♠♦s q✉❡ s❡ x ∈ supp(β) ∩ G ❡♥tã♦ β x = σ(x)β x

  ✱ ♦✉ s❡❥❛ (1 x = 0; − σ(x))β ) x = σ(x)β x s❡ x ∈ supp(β) ∩ (G\G ✱ ❡♥tã♦ β ✳ ❱✐st♦ q✉❡ ❡st❛s sã♦ ❛s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ σ

  ✉♠ ❡❧❡♠❡♥t♦ s❡❥❛ s✐♠étr✐❝♦✱ ❝♦♥❝❧✉í♠♦s ❡♥tã♦ q✉❡ (RG) é ❣❡r❛❞♦ ❝♦♠♦ R✲♠ó❞✉❧♦ ♣❡❧♦s ❝♦♥❥✉♥t♦s 1

  = , α(1 S {αx : x ∈ G ∗ − σ(x)) = 0} ❡ 2

  = : x / S {x + σ(x)x ∈ G ∗ } . ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ♦s ❣❡r❛❞♦r❡s ❞❡ RG sã♦ ❞❛❞♦s ♣♦r 1 = , (1 + σ(x))α = 0

  A {αx : x ∈ G } ❡ 2 = : x / A {x − σ(x)x ∈ G } .

  ◆♦t❡ t❛♠❜é♠ q✉❡ ♦ ❢❛t♦ ❞❡ σ s❡r ❣❡♥❡r❛❧✐③❛❞❛ ♥ã♦ ❢♦✐ ✉t✐❧✐③❛❞♦✱ ♣♦rt❛♥t♦✱ ❡ss❛ ♠❡s♠❛ ❝♦♥str✉çã♦ ❛✐♥❞❛ é ✈á❧✐❞❛ ♥♦ ❝❛s♦ ❞❡ ♦r✐❡♥t❛çõ❡s ✉s✉❛✐s ❡✱ ❞❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ ❡st❡s t❛♠❜é♠ sã♦ ♦s ❣❡r❛❞♦r❡s ❞♦ ❝❛s♦ q✉❡ ♦♠✐t✐♠♦s ♥❛ s❡çã♦ ❛♥t❡r✐♦r✳

  ❊ss❛ ♥♦✈❛ ✐♥✈♦❧✉çã♦ é ♦❜❥❡t♦ r❡❝❡♥t❡ ❞❡ ♣❡sq✉✐s❛ ❡✱ ♣♦rt❛♥t♦✱ ♣♦✉❝♦s r❡s✉❧t❛❞♦s ❛❝❡r❝❛ ❞❡❧❛ sã♦ ❝♦♥❤❡❝✐❞♦s✳ ❙❡❣✉✐♥❞♦ ♦ ❝❛♠✐♥❤♦ ❛♣♦♥t❛❞♦ ♣❡❧♦s ❛rt✐❣♦s ❥á ❝✐t❛❞♦s✱ ♣♦❞❡✲ ♠♦s ❛✜r♠❛r q✉❡✱ ❛té ♦ ♣r❡s❡♥t❡ ♠♦♠❡♥t♦✱ ❛♣❡♥❛s ❢♦✐ r❡❛❧✐③❛❞♦ ♦ ❡st✉❞♦ ❞♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ▲✐❡ ♥♦ ❝❛s♦ ❞❡ ∗ s❡r ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✳ ❚❛✐s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬❱✶✸❪✳

  ◆♦ ✐♥t✉✐t♦ ❞❡ ❡st❡♥❞❡r ♦s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ♣❛r❛ ❡ss❛ ♥♦✈❛ ✐♥✈♦❧✉çã♦✱ ✐r❡♠♦s✱ + ♥♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸✱ ❝❛r❛❝t❡r✐③❛r q✉❛♥❞♦ ♦s ❝♦♥❥✉♥t♦s RG ❡ RG sã♦ ❛♥t✐❝♦♠✉t❛t✐✈♦s✱ + r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡✱ ♥♦ ❈❛♣ít✉❧♦ ✹✱ ✐♥✐❝✐❛r ♦ ❡st✉❞♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ RG ♥♦ ❝❛s♦ ❞❡ σ s❡r ✉♠❛ ♦r✐❡♥t❛çã♦ ✉s✉❛❧ ❡ ∗ ✉♠❛ ✐♥✈♦❧✉çã♦ q✉❛❧q✉❡r✱ ✈✐st♦ q✉❡ ✉♠ r❡s✉❧t❛❞♦ ❞❡ss❡ t✐♣♦ ✐rá ❞❡s❝r❡✈❡r ❛ ❡str✉t✉r❛ ❞♦ s✉❜❣r✉♣♦ C ≤ G✱ ❡ ❡st❡ ♣♦ss✐✈❡❧♠❡♥t❡ s❡rá ❡ss❡♥❝✐❛❧ ♥❛ ❞❡s❝r✐çã♦ ❛♥á❧♦❣❛ q✉❛♥❞♦ σ ❢♦r ✉♠❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛✳

  ❈❛♣ít✉❧♦ ✷

❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❆♥t✐ss✐♠étr✐❝♦s

− −

  = RG ◆❡st❡ ❈❛♣ít✉❧♦ ✐r❡♠♦s ❞❡s❝r❡✈❡r ❛ ❡str✉t✉r❛ ❞♦s ❣r✉♣♦s G t❛✐s q✉❡ RG σ

  é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ s❡♥❞♦ σ ✉♠❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛ ❡ charR 6= 2✳ ❉❡st❛ ❢♦r♠❛✱ ❡st❛s ❞✉❛s ú❧t✐♠❛s ❤✐♣ót❡s❡s s❡rã♦ ♦♠✐t✐❞❛s ❞♦s ❡♥✉♥❝✐❛❞♦s ❞♦s r❡s✉❧t❛❞♦s✳ − −

  P❛r❛ ❡st✉❞❛r RG ✱ ❧❡♠❜r❛♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ RG é ❣❡r❛❞♦ ♣♦r 1 = A {αx : x ∈ G ❡ α(1 + σ(x)) = 0} 2

  = : x A {x − σ(x)x ∈ G\G } . ❖s ❧❡♠❛s ❛ s❡❣✉✐r sã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ tr❛❜❛❧❤♦✱ ♣♦✐s sã♦ ❡❧❡s q✉❡ ❡①✐❜❡♠ ❛s ♣r✐♥❝✐♣❛✐s r❡❧❛çõ❡s ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ ❡♥tr❡ s✐✱ ❡ ❡♥tr❡

  ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ G ❡ ❞♦ ❛♥❡❧ R✳ ▲❡♠❛ ✷✳✶✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❙❡ x ∈ G ∗ ❡♥tã♦ σ(x) 6= −1✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ x ∈ G ❡ σ(x) = −1✱ ❡♥tã♦✱ ♣❛r❛ t♦❞♦ α ∈ R✱ ❛ ❡q✉❛çã♦ α(1+σ(x)) = 0 − −

  é s❛t✐s❢❡✐t❛✱ ♣♦rt❛♥t♦ αx ∈ RG ✳ ❆ss✐♠✱ t♦♠❛♥❞♦ α = 1✱ αx = x ∈ RG ❡✱ ❝♦♠♦ RG é 2 2 =

  ❛♥t✐❝♦♠✉t❛t✐✈♦✱ t❡♠♦s q✉❡ x ❛♥t✐❝♦♠✉t❛ ❝♦♥s✐❣♦ ♠❡s♠♦✱ ♦✉ s❡❥❛ x −x ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ 2 = 0

  ❡♠ 2x ✱ ✉♠ ❛❜s✉r❞♦✱ ❥á q✉❡ charR 6= 2 ❡ RG é ❧✐✈r❡♠❡♥t❡ ❣❡r❛❞♦ ♣♦r G✳ ▲❡♠❛ ✷✳✷✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❙❡ ∗ 6= Id✱ ❡♥tã♦ charR = 4 ❡✱ ♣❛r❛ ∗ ∗ 2 2 ∗ ∗ = x x ) = 0 t♦❞♦ x /∈ G t❡♠✲s❡ xx ✱ x ∈ G ❡ 2(1 + σ(x) ✳ ∗ − − )

  ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ x /∈ G ✳ ❙❛❜❡♠♦s q✉❡ (x − σ(x)x ∈ RG ❡✱ ❝♦♠♦ RG é ❛♥t✐❝♦✲ ♠✉t❛t✐✈♦✱ t❡♠♦s ∗ ∗ ∗ ∗

  (x )(x ) = )(x ), − σ(x)x − σ(x)x −(x − σ(x)x − σ(x)x

  ♣♦rt❛♥t♦ 2 0 = 2(x ) 2 − σ(x)x ∗ ∗ ∗ 2 2

  2 ✷✵ ❈♦♠♦ charR 6= 2✱ ❡♥tã♦ 2 6= 0✱ ♣♦rt❛♥t♦ x ❞❡✈❡ s❡r ✐❣✉❛❧ ❛ ❛❧❣✉♠ ♦✉tr♦ ❡❧❡♠❡♥t♦ ∗ ∗ 2 x

  ❞❡ G q✉❡ ❛♣❛r❡❝❡ ♥❛ ❡q✉❛çã♦✳ ❖❜s❡r✈❡ q✉❡ x 6= x 6= xx ✱ ♣♦✐s x /∈ G ✱ ❛ss✐♠✱ 2 ∗ ∗ 2 2 2 = (x ) = (x )

  ❞❡✈❡♠♦s t❡r ♥❡❝❡ss❛r✐❛♠❡♥t❡ x ✱ ❧♦❣♦ x ∗ ❀ ♣❡❧♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦✱ ∗ ∗ ∈ G x = xx ❡♥❝♦♥tr❛♠♦s q✉❡ x ✳ ❙✉❜st✐t✉✐♥❞♦ ❡ss❛s ✐♥❢♦r♠❛çõ❡s ♥❛ ❡q✉❛çã♦ ❛❝✐♠❛ ♦❜t❡♠♦s 2 2

  2(1 + σ(x) )x = 0, − 4σ(x)xx 2

  ) = 0 = 4σ(x) ♣♦rt❛♥t♦✱ ♣❛r❛ q✉❡ ❛ ♠❡s♠❛ ❢❛ç❛ s❡♥t✐❞♦✱ é ♥❡❝❡ssár✐♦ q✉❡ 2(1 + σ(x) ✳ P♦r ✜♠✱ ❝♦♠♦ 4σ(x) = 0✱ ❡ σ(x) ∈ U(R)✱ t❡♠♦s q✉❡ 4 = 0✱ ♦✉ s❡❥❛✱ charR = 4✳ G

  ❖ s❡❣✉✐♥t❡ ❝♦r♦❧ár✐♦ ❛✜r♠❛ q✉❡✱ s❡ ∗ 6= Id| ✱ ❡♥tã♦ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ G ∗ t❡♠ ❛♦ ♠❡♥♦s ✉♠ ♠ú❧t✐♣❧♦ ♣r❡s❡♥t❡ ♥♦s ❣❡r❛❞♦r❡s ❞❡ RG ✳ G ∗ ❈♦r♦❧ár✐♦ ✷✳✸✳ ❙❡ ♦s ❛♥t✐ss✐♠étr✐❝♦s ❛♥t✐❝♦♠✉t❛♠ ❡ ∗ 6= Id| ✱ ❡♥tã♦✱ ❞❛❞♦ x ∈ G ✱ s❡♠♣r❡ ❡①✐st❡ α ∈ R\ {0} t❛❧ q✉❡ αx ∈ RG ✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ σ(x) = 1✱ ❡♥tã♦ ❜❛st❛ t♦♠❛r α = 2✱ ❥á q✉❡ charR = 4✳ ❙❡ σ(x) 6= 1✱ 2

  ❡♥tã♦ α = 1 − σ(x) 6= 0 ❡ s❛t✐s❢❛③ ❛ ❡q✉❛çã♦✱ ♣♦✐s (1 − σ(x))(1 + σ(x)) = 1 − σ(x) ✱ ❡ 2 = 1

  ❝♦♠♦ x ∈ G ✱ ♣❡❧❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❛ ♦r✐❡♥t❛çã♦ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦✱ t❡♠♦s σ(x) ✳ G ❖❜s❡r✈❡ q✉❡ ♦ ❝♦r♦❧ár✐♦ ❝♦♥t✐♥✉❛ ✈á❧✐❞♦ ♠❡s♠♦ s❡ ∗ = Id| ✱ ♥ã♦ ❣❛r❛♥t✐♥❞♦

  ❛♣❡♥❛s ❛ ❡①✐stê♥❝✐❛ ❞❡ t❛✐s ❡s❝❛❧❛r❡s ♣❛r❛ ♦s ❡❧❡♠❡♥t♦s ❡♠ N✳ ▲❡♠❛ ✷✳✹✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❉❛❞♦s x, y ∈ G✱ ❡♥tã♦✱ xy = yx s❡✱ ❡ ∗ ∗ ∗ ∗ ∗ ∗ y = yx y = y x s♦♠❡♥t❡ s❡✱ x ✳ ❆❧é♠ ❞✐ss♦✱ s❡ x, y /∈ G ∗ ❡ xy = yx✱ ❡♥tã♦ xy = yx = x ∗ ∗ ∗ ∗

  = y x = x y = yx ❡ xy ✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ xy = yx✳ ❙❡ x ∈ G ✱ tr✐✈✐❛❧♠❡♥t❡ t❡♠♦s ❛ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❙❡ y ∈ G ✱ ❡♥tã♦✱ ❛♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦ ❡♠ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❡ xy = yx✱ ♦❜t❡♠♦s ∗ ∗ yx = x y

  ✱ ❡ ♦ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦✳ ∗ ∗ − ), (y ) P♦❞❡♠♦s ❛ss✉♠✐r q✉❡ x, y /∈ G ✳ ◆❡ss❡ ❝❛s♦✱ (x − σ(x)x − σ(y)y ∈ RG ✱

  ❛ss✐♠✱ ∗ ∗ ∗ ∗ (x )(y ) = )(x ),

  − σ(x)x − σ(y)y −(y − σ(y)y − σ(x)x ❧♦❣♦✱ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ xy + yx + σ(xy)x y + σ(xy)y x = σ(y)xy + σ(y)y x + σ(x)yx + σ(x)x y. ∗ ∗ ∗ ∗ ✭✷✳✶✮ x = x y ❝♦♠♦ xy = yx✱ ❛♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦✱ t❡♠♦s y ✱ ❛ss✐♠ ∗ ∗ ∗ ∗ ∗ ∗ 2xy + 2σ(xy)x y = σ(y)xy + σ(y)y x + σ(x)yx + σ(x)x y.

  ❖❜s❡r✈❡ q✉❡ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ ❞♦ ♠❡♠❜r♦ ❞✐r❡✐t♦ ❞❛ ❡q✉❛çã♦ ♣♦❞❡ s❡r ✐❣✉❛❧ ❛

  ✷✶ ♣❛r❛ q✉❡ ❛ ❡q✉❛çã♦ ❢❛ç❛ s❡♥t✐❞♦✱ ❛♠❜♦s ♦s ♠❡♠❜r♦s sã♦ ♥✉❧♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♠♦ 2 6= 0 ∗ ∗ y ❡ σ ∈ U(G)✱ ❡♥tã♦ xy = x ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞♦ ♠❡♠❜r♦ ❞✐r❡✐t♦ é ✐❣✉❛❧ ❛ ❛♣❡♥❛s ✉♠ ♦✉tr♦ ❡❧❡♠❡♥t♦ ❞♦ ♠❡s♠♦ ♠❡♠❜r♦ ♦✉ ♦s q✉❛tr♦ ❡❧❡♠❡♥t♦s sã♦ ✐❣✉❛✐s ❡♥tr❡ s✐✳ ∗ ∗ ∗ ∗ ∗ ∗ y y = y x y = xy

  ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ x ✱ ♦✉ s❡❥❛✱ ♦✉ x ♦✉ x ✳ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 6= yx y = y x = yx y) = yx y = yy x = (y y)x ❙❡ x ✱ ❡♥tã♦ xy ✱ ♣♦rt❛♥t♦✱ x(y ✱ ❛ss✐♠✱ ∗ ∗ ∗ ∗ ∗ ∗ y) ) = 1 y = xy x = yx

  ❝♦♠♦ (x, y) = 1 = (x, y ✱ ❡♥tã♦ (x, y ✱ ❛❜s✉r❞♦❀ s❡ x ✱ ❡♥tã♦ y ✱ ∗ ∗ ∗ ∗ ∗ ∗ ) = yxy = yx y = y xy = (y y)x )

  ♣♦rt❛♥t♦✱ x(yy ✱ ❛ss✐♠✱ ❝♦♠♦ (x, y) = 1 = (x, yy ✱ ∗ ∗ ∗ ) = 1 y = yx

  ❡♥tã♦ (x, y ✱ ❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦✱ x ✳ ❖❜s❡r✈❡ q✉❡ s❡ ♦s q✉❛tr♦ ❡❧❡♠❡♥t♦s ❞♦ ♠❡♠❜r♦ ❞✐r❡✐t♦ ♥ã♦ ❢♦r❡♠ t♦❞♦s ✐❣✉❛✐s ❡♥tr❡ s✐✱ ❡♥tã♦ 2σ(x) = 0✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ ∗ ∗ ∗ ∗

  = y x = x y = yx P♦rt❛♥t♦ xy ✳

  P❛r❛ ❛ r❡❝í♣r♦❝❛✱ ❜❛st❛ ❛♣❧✐❝❛r ♦ r❡s✉❧t❛❞♦ ❛ x ❡ y✳ σ ❊ss❡ ú❧t✐♠♦ ❧❡♠❛ ♥♦s ❛✜r♠❛ q✉❡ s❡ (RG) ∗ é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ ❞♦✐s ❡❧❡♠❡♥t♦s

  ❞❡ G ❝♦♠✉t❛♠ ❡♥tr❡ s✐ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✉♠ ❞❡❧❡s ❝♦♠✉t❛ ❝♦♠ ❛ ✐♠❛❣❡♠ ♣❡❧❛ ✐♥✈♦❧✉çã♦ ❞♦ ♦✉tr♦✳ ❊ss❡ ❢❛t♦ ❡✈❡♥t✉❛❧♠❡♥t❡ s❡rá ✉t✐❧✐③❛❞♦ s❡♠ ❢❛③❡r r❡❢❡rê♥❝✐❛ ❛♦ ❧❡♠❛✳ ∗ ∗

  ❆ ♣❛rt✐r ❞❡ ❛❣♦r❛ ✐r❡♠♦s ♣❛rt✐❝✐♦♥❛r G ❡♠ G ❡ G\G ❡ ♣r♦❝✉r❛r ❡①✐❜✐r ♦ q✉❡ ♦❝♦rr❡ ❝♦♠ ♦ ♣r♦❞✉t♦ ❞❡ ❡❧❡♠❡♥t♦s ❡♥tr❡ ❡st❡s s✉❜❝♦♥❥✉♥t♦s✳ ▼♦str❛r❡♠♦s ♥♦s ❧❡♠❛s ❛ s❡❣✉✐r q✉❡ ❛ ✐♥✈♦❧✉çã♦ ❡♠ ✉♠ ❡❧❡♠❡♥t♦ ♥ã♦ s✐♠étr✐❝♦ x é ❞❛❞❛ ♣❡❧❛ ❝♦♥❥✉❣❛çã♦ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ y q✉❛❧q✉❡r ❞♦ ❣r✉♣♦ G t❛❧ q✉❡ xy 6= yx✳ ▼♦str❛r❡♠♦s t❛♠❜é♠ ❛ r❡❧❛çã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ♦ ❝♦♠✉t❛❞♦r ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s x ❡ y ❞❡st❡s s✉❜❝♦♥❥✉♥t♦s ❡ ❛ s✐♠❡tr✐❛ ❞♦ ♣r♦❞✉t♦ xy✳

  Pr✐♠❡✐r♦✱ ✈❛♠♦s ❡st✉❞❛r ♦ q✉❡ ♦❝♦rr❡ ❝♦♠ ♦s ❡❧❡♠❡♥t♦s ❡♠ G\G ✳ ∗ ∗ ▲❡♠❛ ✷✳✺✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❙❡ x /∈ G ❡ y /∈ N ∪ G ✱ ❡♥tã♦✱ y x , x

  ∈ {x }✳ ∗ ∗ − ), (y )

  ❉❡♠♦♥str❛çã♦✳ P♦r ❤✐♣ót❡s❡ t❡♠♦s (x − σ(x)x − σ(y)y ∈ RG ✱ ❛ss✐♠ ∗ ∗ ∗ ∗ (x )(y ) = )(x ),

  − σ(x)x − σ(y)y −(y − σ(y)y − σ(x)x ❧♦❣♦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ xy + yx + σ(xy)x y + σ(xy)y x = σ(y)xy + σ(y)y x + σ(x)yx + σ(x)x y. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ✭✷✳✷✮ y , y x , xy , y x, x y, yx

  ❈♦♠♦ ❡st❛ ❡q✉❛çã♦ é ✈❡r❞❛❞❡✐r❛✱ ❡♥tã♦ xy ∈ {yx, x }✳ ∗ ∗ , x y

  ◆♦t❡ q✉❡ xy /∈ {xy }✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ x ♦✉ y ∈ G ✱ ❝♦♥tr❛r✐❛♥❞♦ ❛ ❤✐♣ót❡s❡✳ ❙❡ ∗ ∗ y xy ∈ {yx, yx }✱ ❡♥tã♦ x ∈ {x, x } ❡ ♦ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦✳ ∗ ∗ ∗ ∗ ∗ ∗ x, x y , y x

  ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦ q✉❡ xy /∈ {yx, yx }✱ ❛ss✐♠✱ t❡♠♦s xy ∈ {y }✳ ❱❛♠♦s ❛♥❛❧✐s❛r ❡st❛s ✸ ♣♦ss✐❜✐❧✐❞❛❞❡s✳ x

  ❈❛s♦ ✶✿ xy = y ✳ ❈♦♠♦ y /∈ N✱ ❞❡✈❡♠♦s t❡r ❛♦ ♠❡♥♦s ✉♠ ♦✉tr♦ t❡r♠♦ ❡♠ ∗ ∗

  ∗ ∗ ∗ ∗ ∗ ∗ ✷✷ y x = xy = y x x = x y ∗ ∗ ∗ ✱ ♦ q✉❡ ✐♠♣❧✐❝❛r✐❛ ❡♠ x ∈ G ✳ ❆♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦ ❡♠ xy = y ✱ x = x y = yx ❡♥❝♦♥tr❛♠♦s y ✱ ❛ss✐♠✱ s✉❜st✐t✉✐♥❞♦ ♥❛ ❡q✉❛çã♦ ✭✷✳✷✮✱ t❡♠♦s ∗ ∗

  (1 + σ(xy) = 0,

  − σ(y))xy + (1 + σ(xy) − σ(x))yx − σ(y)xy − σ(x)yx ∗ ∗ ∗ = yx s❡♥❞♦ q✉❡ ♣❛r❛ ❡st❛ ❢❛③❡r s❡♥t✐❞♦✱ é ♥❡❝❡ssár✐♦ q✉❡ xy ✱ ❧♦❣♦ xy ∈ G ✱ ❡ σ(x) = ∗ −1 ∗

  ) = σ(y ) ) = −σ(y)❀ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❡ss❛ ú❧t✐♠❛ ❡q✉❛çã♦ ♣♦r σ(y ✱ ♦❜t❡♠♦s σ(xy −1✱ ❝♦♥tr❛r✐❛♥❞♦ ♦ ▲❡♠❛ ✷✳✶✳ ∗ ∗ y

  ❈❛s♦ ✷✳❛✿ xy = x ❡ σ(xy) 6= −1✳ ❙❛❜❡♠♦s q✉❡ xy ❞❡✈❡ s❡r ✐❣✉❛❧ ❛ ♦✉tr♦ x ❡❧❡♠❡♥t♦ ❞❛ ❡①♣r❡ssã♦✳ ❙❡ xy = y ✱ ♣♦❞❡♠♦s ♣r♦❝❡❞❡r ❝♦♠♦ ♥♦ ❝❛s♦ ❛♥t❡r✐♦r✳ ❉❡ss❛ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ y = y x y = y x ❢♦r♠❛✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ xy = x ✳ ❆♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦ ❡♠ x ✱ ❡♥❝♦♥tr❛♠♦s xy = yx✱ ❝♦♥tr❛❞✐çã♦✳ ∗ ∗ ∗ ∗ y x = yx

  ❈❛s♦ ✷✳❜✿ xy = x ❡ σ(xy) = −1✳ ❆♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦✱ t❡♠♦s y ✱ ∗ ∗ − − = yx = xy + yx

  ❧♦❣♦ (xy) ✱ ❛ss✐♠ xy − σ(xy)(xy) ∈ RG ✱ ♣♦rt❛♥t♦✱ ❝♦♠♦ RG é −1 ∗ −∗ ) = x

  ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡ ❞❡♥♦t❛♥❞♦ (x ✱ t❡♠♦s −1 −1 −∗ −1 −1 −∗ (xy + yx)(x )x ) = )x )(xy + yx),

  − σ(x −(x − σ(x ❧♦❣♦✱ −1 −1 xyx + y + y + x yx −1 −∗ −1 −∗ −1 −∗ −1 −∗ =

  σ(x )xyx + σ(x )yxx + σ(x )x xy + σ(x )x yx, −1 −∗ −∗ −1 −∗ −∗ , xyx , yxx , x yx, x xy, x yx

  ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ y ∈ {xyx }✳ ❖❜s❡r✈❡ q✉❡ s❡ y ∈ −1 −∗ −1 −∗ −∗ y , xyx , x yx, yxx , x xy

  {xyx }✱ ❡♥tã♦ ❡♥❝♦♥tr❛r❡♠♦s x ∈ G ♦✉ x ∈ {x, x }✱ ❝♦♥✲ ∗ −1 ∗ ∗ ∗ ∗ yx y x y tr❛❞✐çã♦❀ ♣♦rt❛♥t♦✱ y = x ✱ ♦✉ s❡❥❛✱ yx = x ✳ ❈♦♠♦ yx = y ✱ s❡ yx = x ✱ ❡♥tã♦✱ ∗ ∗ ∗ y x = x y x

  ✱ ❡ ❛♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦✱ ❡♥❝♦♥tr❛♠♦s xy = y ✱ ❞♦♥❞❡✱ ♣❡❧♦ ❈❛s♦ ✶✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ❛ ❤✐♣ót❡s❡✳ ∗ ∗ x

  ❈❛s♦ ✸✿ xy = y ✳ P❡❧♦s ❝❛s♦s ❛♥t❡r✐♦r❡s✱ é s✉✜❝✐❡♥t❡ ❛♥❛❧✐s❛r ♦ ❝❛s♦ ❡♠ q✉❡ xy é ❞✐❢❡r❡♥t❡ ❞♦s ❞❡♠❛✐s ❡❧❡♠❡♥t♦s ❞❛ ❡q✉❛çã♦✳ ◆❡ss❡ ❝❛s♦✱ t❡♠♦s q✉❡ σ(xy) = −1✱ ❛❧é♠ ❞♦ ♠❛✐s xy ∈ G ✱ ♦ q✉❡ s❛❜❡♠♦s✱ ♣❡❧♦ ▲❡♠❛ ✷✳✶✱ s❡r ✐♠♣♦ssí✈❡❧✳ y

  ❆ss✐♠✱ ♣♦r t♦❞♦s ♦s ❝❛s♦s ❛♥❛❧✐③❛❞♦s✱ ❡♥❝♦♥tr❛♠♦s q✉❡✱ x ∈ {x, x }✳ ❈♦♠♦ ❢♦✐ ❞✐t♦ ♥♦ ❈❛♣ít✉❧♦ ✶✱ ♣❛r❛ ❡st✉❞❛r ❛s ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s✱ é ♥❡❝❡ssár✐♦

  ❝♦♥❤❡❝❡r ❞❡ ❛♥t❡♠ã♦ ♦s r❡s✉❧t❛❞♦s ❛❝❡r❝❛ ❞❛s ✐♥✈♦❧✉çõ❡s ✐♥❞✉③✐❞❛s✳ ◆❡st❡ ❝❛s♦✱ ❛ ❞❡s❝r✐çã♦ ❞❡ q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ RG ∗ é ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡ ❡♥❝♦♥tr❛ ❡♠ ❬●P✶✸❛❪ ❡ é ❞❛❞♦ ♣♦r✿ ❚❡♦r❡♠❛ ✷✳✻ ✭❚❡♦r❡♠❛ ✷✳✷✱ ❬●P✶✸❛❪✮✳ ❙❡ charR 6= 2✱ ❡♥tã♦ RG ∗ é ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦❝♦rr❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

  ✭❆✮ G é ❛❜❡❧✐❛♥♦ ❡ ∗ = Id✳

   ✷✸ ✭❈✮ charR = 4✱ G é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s ❡ g ∈

  {g, sg} , ∀g ∈ G✳ ❉❡ss❛ ❢♦r♠❛✱ ❛♦ ❡st✉❞❛r♠♦s ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ (RG) σ ✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡

  ❞❡✈❡♠♦s t❡r q✉❡ N s❛t✐s❢❛③ ❛❧❣✉♠❛ ❞❡ss❛s ❝♦♥❞✐çõ❡s✳ ▲❡♠❛ ✷✳✼✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❙❡ x, y /∈ G ✱ ❡♥tã♦✱ sã♦ ✈❡r❞❛❞❡✐r❛s✿ y ∗ ∗ ∗ y

  ✭✐✮ x ∈ {x, x }✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ xy = x ✳ ✭✐✐✮ xy ∈ G ⇔ xy = yx✳

  ✭✐✐✐✮ ❙❡ xy 6= yx✱ ❡♥tã♦ 1 + σ(xy) = σ(x) + σ(y)✳ ❙❡ xy = yx✱ ❡♥tã♦ 2(1 + σ(xy)) = 0 = 2(σ(x) + σ(y))

  ✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ (x, y) = 1✱ ❡♥tã♦ ♦ ✐t❡♠ ✭✐✮ s❡❣✉❡ ❞♦ ▲❡♠❛ ✷✳✹✳ ❉❡ss❛ ❢♦r♠❛✱ s✉♣♦♥❤❛ q✉❡ (x, y) 6= 1✳

  ❙❡ x, y ∈ N✱ ❡♥tã♦✱ N s❛t✐s❢❛③ ♦ ✐t❡♠ ✭❈✮ ❞♦ ❚❡♦r❡♠❛ ✷✳✻ ❡ ♣♦ss✉✐ ✉♠ ú♥✐❝♦ y = x(x, y) = xs = x

  ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s✱ ❧♦❣♦ x ✳ ❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ xy = xssy = ∗ ∗ x y ✳

  ❙❡ x, y /∈ N✱ ❡♥tã♦✱ ❛♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✷✳✺ ❛ x ❡ y✱ t❡♠♦s q✉❡ xy = yx ✱ ❛❧é♠ ❞♦ ∗ ∗ ∗ ∗ ∗ ∗ = x y y

  ♠❛✐s✱ ❛♣❧✐❝❛♥❞♦ ♦ ♠❡s♠♦ ❧❡♠❛ ❛ y ❡ x ✱ ❡♥❝♦♥tr❛♠♦s q✉❡ yx ✱ ♦✉ s❡❥❛ xy = x ✳ y ∗ = x

  ❙❡ x ∈ N ❡ y /∈ N✱ ♣❡❧♦ ▲❡♠❛ ✷✳✺✱ t❡♠♦s x ✳ ❖❜s❡r✈❡ q✉❡ ♦ ♠❡s♠♦ ∗ ∗ ∗ ∗ ∗ −1 y = y x r❡s✉❧t❛❞♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ ❛ x ❡ y ✱ ♦✉ s❡❥❛✱ x ✳ ◆♦t❡ q✉❡ x ∈ N\G ❡ −1 −1 −1 ∗ −1 ∗ ∗ −1 ∗ −1 (x , y) = (x ) y ) y = (yx ) /

  6= 1✱ ♣♦rt❛♥t♦✱ yx 6= (x ✱ ♦✉ s❡❥❛ yx ∈ G ✱ ❛ss✐♠✱ −1 −1 ∗ x yx −1 −1 / = y = y

  ❛♣❧✐❝❛♥❞♦ ♦ ❝❛s♦ ❛♥t❡r✐♦r ❛ y ❡ yx ✱ ❥á q✉❡ yx ∈ N✱ ♦❜t❡♠♦s y ✱ ♦✉ ∗ ∗ ∗ x = x y s❡❥❛✱ xy = y ✳ ∗ ∗ ∗

  = y x ❙❡ x /∈ N ❡ y ∈ N✱ ♣❡❧♦ ❝❛s♦ ❛♥t❡r✐♦r✱ t❡♠♦s q✉❡ yx = xy ✳ ❆♣❧✐❝❛♥❞♦ ❛ ∗ ∗ ∗ ∗ ∗ y

  = y x = xy = x ✐♥✈♦❧✉çã♦ ❡♠ xy ✱ ❡♥❝♦♥tr❛♠♦s yx ✱ ♦✉ s❡❥❛✱ x ✳ ❆❧é♠ ❞✐ss♦✱ ❛♣❧✐❝❛♥❞♦ ∗ ∗ ∗ ∗ x y ❛ ✐♥✈♦❧✉çã♦ ❡♠ yx = y ✱ ✈❡r✐✜❝❛♠♦s xy = x ✳ ❉❡ss❛ ❢♦r♠❛✱ ❣❛r❛♥t✐♠♦s ✭✐✮✳

  P❛r❛ ✈❡r✐✜❝❛r ✭✐✐✮ ♦❜s❡r✈❡ q✉❡✱ ∗ ∗ ∗ ∗ xy y = xy = y x ∈ G ⇔ x ✭✐t❡♠ ✭✐✮✮ ∗ ∗ x = xy

  ⇔ yx = y ✭❛♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦✮. ❱❛♠♦s ♠♦str❛r ✭✐✐✐✮✳ ❈♦♠♦ x, y /∈ G ❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ t❡♠♦s ∗ ∗ ∗ ∗

  (x )(y ) = )(x ), − σ(x)x − σ(y)y −(y − σ(y)y − σ(x)x

  ♦✉ s❡❥❛

  ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ✷✹ y x = x y x = yx P❡❧♦ ✐t❡♠ ✭✐✮✱ xy = x ✱ yx = y ✱ xy ❡ y ✱ ♣♦rt❛♥t♦ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ s❡ r❡❡s❝r❡✈❡ ❝♦♠♦ ∗ ∗ (1 + σ(xy))xy + (1 + σ(xy))yx = (σ(x) + σ(y))x y + (σ(x) + σ(y))yx . ∗ ∗ y

  ❆ss✐♠✱ s❡ xy 6= yx✱ ♥♦✈❛♠❡♥t❡ ♣❡❧♦ ✐t❡♠ ✭✐✮✱ xy = yx ✱ ❛❧é♠ ❞✐ss♦✱ xy 6= x ✱ ❥á q✉❡ x /∈ G ✱ ♣♦rt❛♥t♦✱ 1 + σ(xy) = σ(x) + σ(y)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ xy = yx✱ ❡♥tã♦ ∗ ∗ ∗ ∗ xy / y, yx y = yx

  ∈ {x }✱ ❥á q✉❡ x / ∈ G ❀ ❛❧é♠ ❞✐ss♦ xy = yx ✐♠♣❧✐❝❛ x ✱ ♣♦rt❛♥t♦✱ ♣❛r❛ q✉❡ ❛ ❡q✉❛çã♦ ❢❛ç❛ s❡♥t✐❞♦✱ ❞❡✈❡♠♦s t❡r q✉❡ 2(1 + σ(xy)) = 0 = 2(σ(x) + σ(y))✳ ❖ ❧❡♠❛ ❛ s❡❣✉✐r ♠♦str❛ ❛ r❡❧❛çã♦ ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ G ❝♦♠ ✉♠ ❡❧❡♠❡♥t♦ ❞❡

  G \G ∗ ∗

  ▲❡♠❛ ✷✳✽✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ y ∈ G ✱ x /∈ G ❡ α t❛❧ q✉❡ αy ∈ RG ✱ sã♦ ✈❡r❞❛❞❡✐r❛s y ∗ ✭✐✮ x ∈ {x, x }✳

  ✭✐✐✮ xy ∈ G ∗ ⇔ xy 6= yx✳ 2

  ✭✐✐✐✮ ❙❡ xy 6= yx✱ ❡♥tã♦ α(1 − σ(x)) = 0✳ ❙❡ xy = yx✱ ❡♥tã♦ α ∈ R ✳ ❉❡♠♦♥str❛çã♦✳ P❡❧♦ ❈♦r♦❧ár✐♦ ✷✳✸✱ ♣♦❞❡♠♦s t♦♠❛r α ♥ã♦ ♥✉❧♦ ❡♠ R t❛❧ q✉❡ αy ∈ RG ✱

  ) ❛ss✐♠✱ ♣♦r ❤✐♣ót❡s❡✱ αy ❛♥t✐❝♦♠✉t❛ ❝♦♠ (x − σ(x)x ✱ ♦✉ s❡❥❛ ∗ ∗ 0 = αy(x ) + (x )αy

  − σ(x)x − σ(x)x ∗ ∗ = α(xy + yx y ).

  − σ(x)x − σ(x)yx ❈♦♠♦ α é ♥ã♦ ♥✉❧♦✱ ♣❛r❛ q✉❡ ❡st❛ ❡q✉❛çã♦ s❡❥❛ ✈❡r❞❛❞❡✐r❛✱ é ♥❡❝❡ssár✐♦ q✉❡ ♦✉ xy = yx✱ ♦✉ xy = yx ✱ ❥á q✉❡ x /∈ G ✳ ■ss♦ ♣r♦✈❛ ✭✐✮✳

  ❆✐♥❞❛ ❛❞♠✐t✐♥❞♦ q✉❡ α 6= 0✱ s❡ ❛ ♣r✐♠❡✐r❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❛♣r❡s❡♥t❛❞❛ ♥♦ ♣❛rá❣r❛❢♦ 2 ❛♥t❡r✐♦r ♦❝♦rr❡✱ ❡♥tã♦ 2α ❞❡✈❡ s❡r ♥✉❧♦ ❡✱ ♣♦rt❛♥t♦✱ α ∈ R ❀ ❝❛s♦ xy = yx ✱ ❡♥tã♦ ❛ ❡q✉❛çã♦ ❢♦r♥❡❝❡ α(1 − σ(x)) = 0✳ ❉❡ss❛ ❢♦r♠❛ ❝♦♥❝❧✉í♠♦s q✉❡ ✭✐✐✐✮ ❞❡❝♦rr❡ ❞♦ ❛r❣✉♠❡♥t♦ ❛❝✐♠❛ ❡ ♦ ❢❛t♦ ❞❡ q✉❡ s❡ α = 0 ❡♥tã♦ tr✐✈✐❛❧♠❡♥t❡ sã♦ ✈❡r✐✜❝❛❞❛ ❛s ❝♦♥❞✐çõ❡s ❞❡ss❡ ✐t❡♠✳

  ❘❡st❛✱ ♣♦rt❛♥t♦✱ ✈❡r✐✜❝❛r ✭✐✐✮✳ P❛r❛ ✐ss♦✱ é s✉✜❝✐❡♥t❡ ♥♦t❛r q✉❡ xy ∗ .

  6= yx ⇔ xy = yx ⇔ xy ∈ G ❆♥t❡s ❞❡ ♣r♦ss❡❣✉✐r ♥♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛s r❡❧❛çõ❡s ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ G✱ ✐r❡♠♦s

  ✈❡r✐✜❝❛r q✉❡✱ s❡ ♦s s✐♠étr✐❝♦s ❞♦ ❣r✉♣♦ ❝♦♠✉t❛♠ ❡♥tr❡ s✐✱ ❡♥tã♦ ♣♦❞❡♠♦s✱ ♥❡st❡ ♠♦♠❡♥t♦✱ ❝❛r❛❝t❡r✐③❛r ♦s ❣r✉♣♦s t❛✐s q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ♠♦str❛♥❞♦ q✉❡✱ ♥❡st❡ ❝❛s♦✱ G é

  ✷✺ ❚❡♦r❡♠❛ ✷✳✾ ✭❚❡♦r❡♠❛ ✷✳✹✱ ❬❏❘✵✻❪✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ♥ã♦ ❛❜❡❧✐❛♥♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ∗ ❡ R

  ✉♠ ❛♥❡❧ t❛❧ q✉❡ charR 6= 2✳ ❆s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿ ✭✶✮ (RG) é ❝♦♠✉t❛t✐✈♦❀ ✭✷✮ G é ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ∗❀ 2 2 = x = x ∗ ∗ y

  ✭✸✮ G/Z(G) ≃ C × C ✱ x s❡ x ∈ Z(G) ❡ x ♣❛r❛ t♦❞♦ y t❛❧ q✉❡ (x, y) 6= 1✳ − ′ ) = Pr♦♣♦s✐çã♦ ✷✳✶✵✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❙❡ (G {1}✱ ❡♥tã♦ G é ❛❜❡❧✐❛♥♦ ♦✉ ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ∗✳ ∗ ∗ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ x ∈ G ❡ y /∈ G ✳ ❙❡ xy 6= yx✱ ❡♥tã♦✱ ♣❡❧♦ ✐t❡♠ ✭✐✐✮ ❞♦ ▲❡♠❛ ✷✳✽✱ t❡♠♦s q✉❡ xy ∈ G ∗ ✱ ❧♦❣♦✱ ♣♦r ❤✐♣ót❡s❡✱ (x, xy) = 1✱ ♦✉ s❡❥❛ x(xy) = (xy)x✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ xy = yx✱ ✉♠ ❛❜s✉r❞♦❀ ♣♦rt❛♥t♦ ❞❡✈❡♠♦s t❡r q✉❡ xy = yx ♣❛r❛ t♦❞♦ x ∈ G ∗ ❡ y /∈ G ∗ ✳

  ) = 1 ❈♦♠♦ ♣♦r ❤✐♣ót❡s❡ (G ∗ ✱ ❝♦♥❝❧✉í♠♦s q✉❡ G ∗ ⊂ Z(G)✳

  ❆✜r♠❛çã♦✿ (RG) é ❝♦♠✉t❛t✐✈♦✳ ∗ ∗ : x / ∗ ◆♦t❡ q✉❡ (RG) é ❣❡r❛❞♦ ♣♦r G ∪{x + x ∈ G }✱ ❧♦❣♦✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡

  ♦s ❡❧❡♠❡♥t♦s ❣❡r❛❞♦r❡s ❝♦♠✉t❛♠ ❡♥tr❡ s✐✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ G ⊂ Z(G)✱ ❜❛st❛ ✈❡r✐✜❝❛r ∗ ∗ ) ) q✉❡ (x + x ❝♦♠✉t❛ ❝♦♠ (y + y ✱ ∀x, y /∈ G ✳ ∗ ∗ ∗ ∗

  )(y + y ) = (y + y )(x + x ) ❙❡ xy = yx ❢❛❝✐❧♠❡♥t❡ ❡♥❝♦♥tr❛♠♦s (x + x ✳ ❙✉♣♦♥❤❛

  ❡♥tã♦ q✉❡ xy 6= yx✱ ❛ss✐♠✱ ∗ ∗ ∗ ∗ ∗ ∗ (x + x )(y + y ) = xy + xy + x y + x y ∗ ∗ ∗ ∗

  = yx + y x + yx + y x ∗ ∗ ✭♣❡❧♦ ✐t❡♠ ✭✐✮ ❞♦ ▲❡♠❛ ✷✳✼✮ = (y + y )(x + x ). ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥❝❧✉í♠♦s ❛ ❛✜r♠❛çã♦✳

  P♦r ✜♠✱ ❝♦♠♦ (RG) é ❝♦♠✉t❛t✐✈♦✱ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✷✳✾✱ ❡♥❝♦♥tr❛♠♦s q✉❡ G

  é ❛❜❡❧✐❛♥♦ ♦✉ ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ∗✳ ❆té ♦ ♠♦♠❡♥t♦ ✈❡r✐✜❝❛♠♦s ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ♦ ♣r♦❞✉t♦ xy q✉❛♥❞♦ ❛♠❜♦s x, y ∗ ♦✉ ✉♠ ❞❡❧❡s ♣❡rt❡♥❝❡ ❛ ❡st❡ ❝♦♥❥✉♥t♦ ❡ ♦ ♦✉tr♦ ❛♦ s❡✉ ❝♦♠♣❧❡♠❡♥t❛r ❡♠

  ∈ G\G G

  ✳ ❆ s❡❣✉✐♥t❡ ♦❜s❡r✈❛çã♦ é ✉♠ ❢❛t♦ ❣❡r❛❧ ❞❡ ✐♥✈♦❧✉çõ❡s ❡♠ ❣r✉♣♦s ❡ s❡rá ✉t✐❧✐③❛❞❛ ♥♦s ❞❡♠❛✐s ❝❛♣ít✉❧♦s✳ ∗ ∗ =

  ❖❜s❡r✈❛çã♦ ✷✳✶✶✳ ❈❛s♦ x, y ∈ G ✱ ♥♦t❡ q✉❡ xy ∈ G s❡✱ ❡ s♦♠❡♥t❡ s❡✱ xy = (xy) ∗ ∗ y x = yx ✳

  ▲❡♠❛ ✷✳✶✷✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ x, y ∈ G ∗ ✳ ❙❡ α, β ∈ R\ {0} sã♦ t❛✐s q✉❡ αx, βy ∈ RG ✱ ❡♥tã♦ xy 6= yx ✐♠♣❧✐❝❛ ❡♠ αβ = 0 ❡ xy = yx ✐♠♣❧✐❝❛ ❡♠ 2αβ = 0✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ αβxy = −βαyx✱ ❛ss✐♠✱ s❡ xy 6= yx ❡ss❛ ❡q✉❛çã♦ só ❢❛③ s❡♥t✐❞♦ s❡ αβ = 0✳ ❙❡ xy = yx✱ ❡♥tã♦ 2αβxy = 0✱ ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s

  ∗ −1 ∗ ✷✻ = xc x x = x x

  ❉❛❞♦ x ∈ G✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r x ✱ s❡♥❞♦ c ♦ ♠✉❧t✐♣❧✐❝❛❞♦r ❞❛ x = 1 ✐♥✈♦❧✉çã♦ ❞♦ ❡❧❡♠❡♥t♦ x✳ ◆❛t✉r❛❧♠❡♥t❡ s❡ x ∈ G ✱ ❡♥tã♦ c ✳ ■r❡♠♦s ♠♦str❛r ♥♦ x ♣ró①✐♠♦ ❧❡♠❛ q✉❡ s❡ x /∈ G ∗ ✱ ❡♥tã♦ c é ✉♠❛ ❝♦♥st❛♥t❡✱ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ s✱ ♣❛r❛ x = c y t♦❞♦ ❡❧❡♠❡♥t♦ ♥ã♦ s✐♠étr✐❝♦✱ ♦✉ s❡❥❛ c ♣❛r❛ t♦❞♦ y /∈ G ∗ ✳ ❆❧é♠ ❞✐ss♦✱ ♠♦str❛r❡♠♦s q✉❡ s❡ G é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ G s❡rá ✉♠ ❣r✉♣♦ ❝í❝❧✐❝♦ ❞❡ 2 ❡❧❡♠❡♥t♦s ❣❡r❛❞♦ ♣♦r ❡ss❛

  ❝♦♥st❛♥t❡ s✳ ▲❡♠❛ ✷✳✶✸✳ ❙❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ 2

  = 1 ✭✐✮ ❙❡ ∗ 6= Id✱ ❡♥tã♦ ❡①✐st❡ s ∈ G t❛❧ q✉❡ s ❡ x ∈ {x, sx = xs} ∀x ∈ G✳

  ✭✐✐✮ ❙❡ G é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ G ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s ∈ G ✭❧♦❣♦ ❞❡ ♦r❞❡♠ 2✮ ❡ x ∈ {x, sx = xs} ∀x ∈ G✳ ❆❧é♠ ❞✐ss♦✱ s ∈ Z(G) ∩ G ✳ −1 ∗ ∗ ∗ x

  ❉❡♠♦♥str❛çã♦✳ ✭✐✮ ❙✉♣♦♥❤❛ q✉❡ ∗ 6= Id✱ ♣♦rt❛♥t♦ G\G 6= ∅✳ ❙❡❥❛♠ x / ∈ G ❡ s = x ✳ ∗ ∗ ∗ = xs ) = 1 = sx

  ◆❛t✉r❛❧♠❡♥t❡ x ❡ ❝♦♠♦ (x, x ✱ ❡♥tã♦ x ❀ ❛❧é♠ ❞✐ss♦✱ ♣❡❧♦ ▲❡♠❛ ✷✳✷✱ 2 −1 ∗ −2 ∗ −2 −∗ 2 2 2 s = (x x ) = x (x ) = x x = 1 ❡ s = xx ✳ P❛r❛ ✜♥❛❧✐③❛r✱ s❡ y /∈ G ✱ ❡♥tã♦✱ ∗ ∗ −∗ −1 ∗ y = y y

  ♣❡❧♦ ✐t❡♠ ✭✐✮ ❞♦ ▲❡♠❛ ✷✳✼✱ t❡♠♦s q✉❡ xy = x ✱ ♦✉ s❡❥❛ s = xx ❀ ❛ss✐♠ s ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ x ♣♦rt❛♥t♦ y

  ∈ {y, sy = ys} ∀y ∈ G✳ ✭✐✐✮ ❙❡ G é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ ❛ ✐❞❡♥t✐❞❛❞❡ ♥ã♦ ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ G✱ ∗ = xs = sx

  ♣♦rt❛♥t♦ Id 6= ∗✱ ❛ss✐♠✱ ♣❡❧♦ ✐t❡♠ ✭✐✮✱ t❡♠♦s q✉❡ s❡ x /∈ G ✱ ❡♥tã♦ x ✳ ❖❜s❡r✈❡ q✉❡ s❡ x /∈ G ❡ y ∈ G sã♦ t❛✐s q✉❡ (x, y) 6= 1✱ ♣❡❧♦ ✐t❡♠ ✭✐✮ ❞♦ ▲❡♠❛ ∗ = x y

  ✷✳✼✱ s❡ y /∈ G ✱ ♦✉ ♣❡❧♦ ✐t❡♠ ✭✐✮ ❞♦ ▲❡♠❛ ✷✳✽✱ s❡ y ∈ G ✱ s❛❜❡♠♦s q✉❡ x ✱ ♦✉ s❡❥❛✱ −1 −1 ∗ −1 y (x, y) = x x = x x = x xs = s

  ❙✉♣♦♥❤❛ q✉❡ x, y ∈ G ❡ (x, y) 6= 1✳ P❡❧♦ ✐t❡♠ ✭✐✮ ❞♦ ▲❡♠❛ ✷✳✶✷✱ t❡♠♦s q✉❡ ∗ ∗ ∗ xy / x = (xy) = sxy

  ∈ G ✱ ❛ss✐♠✱ ♣❡❧♦ ✐t❡♠ ✭✐✮✱ t❡♠♦s q✉❡ yx = y ✱ ♦✉ s❡❥❛ (x, y) = s✳ −1 sx = (s, x) = s P♦r ✜♠✱ ♥♦t❡ q✉❡ s❡ (s, x) 6= 1✱ ♣❛r❛ ❛❧❣✉♠ x ∈ G✱ ❡♥tã♦ sx ✱ x 2

  = 1 = s = 1 ♦✉ s❡❥❛ s ✱ ❛❜s✉r❞♦✱ ♣♦rt❛♥t♦✱ s ∈ Z(G)✳ ❆❧é♠ ❞✐ss♦✱ s❡ s /∈ G ✱ ❡♥tã♦ s ✱

  = 1 ❛❜s✉r❞♦✱ ♣♦✐s 1 6= s✳ ∗ ∗ ∗

  = y x = yx ◆♦t❡ q✉❡ s❡ ∗ = Id✱ ❡♥tã♦ ♣❛r❛ t♦❞♦ x, y ∈ G✱ xy = (xy) ❡✱

  ♣♦rt❛♥t♦✱ G é ❛❜❡❧✐❛♥♦✳ ❈♦♠ ❡ss❛ ✐♥❢♦r♠❛çã♦✱ ❡st❛♠♦ ♣r♦♥t♦s ♣❛r❛ ♣r♦✈❛r ♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✳ ❚❡♦r❡♠❛ ✷✳✶✹✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ charR 6= 2✱ G ✉♠ ❣r✉♣♦ ❝♦♠ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❡ σ : G → U(R) ✉♠❛ ♦r✐❡♥t❛çã♦ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ∗ ❡ N = kerσ✳ ❊♥tã♦✱ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ✈❡r❞❛❞❡✐r❛s✿

  ✭✐✮ ❖❝♦rr❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ♣♦ss✐❜✐❧✐❞❛❞❡s✿ G ✭❛✮ ∗ = Id ❡ G é ❛❜❡❧✐❛♥♦✳

   ✷✼ ✭✶✮ G é ❛❜❡❧✐❛♥♦ ❡ ❡①✐st❡ s ∈ G ∩ Z(G)✱ t❛❧ q✉❡ x ∈ {x, xs} , ∀x ∈ G✳ ′ ∗

  = ∗ ✭✷✮ G {1, s} ⊂ N ∩ Z(G) ❡ x ∈ {x, xs} , ∀x ∈ G✳

  ✭✐✐✮ ∀x, y /∈ G ∗ ✱ s❡ xy 6= yx✱ ❡♥tã♦ 1 + σ(xy) = σ(x) + σ(y)❀ s❡ xy = yx✱ ❡♥tã♦ 1 + σ(xy) 2 ❡ σ(x) + σ(y) ∈ R ✳ ∗ ∗ , y

  ✭✐✐✐✮ ∀x /∈ G ∈ G ❡ α ∈ R t❛❧ q✉❡ αy ∈ RG ✱ s❡ xy 6= yx✱ ❡♥tã♦ α(1 − σ(x)) = 0❀ 2 s❡ xy = yx✱ ❡♥tã♦ α ∈ R ✳ ✭✐✈✮ ∀x, y ∈ G ❡ α, β ∈ R t❛✐s q✉❡ αx, βy ∈ RG ✳ ❙❡ xy 6= yx✱ ❡♥tã♦ αβ = 0❀ s❡ 2 xy = yx

  ✱ ❡♥tã♦ αβ ∈ R ✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ RG s❡❥❛ ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❙❡ ∗ = Id✱ ❥á ♠♦str❛♠♦s q✉❡ G é ❛❜❡❧✐❛♥♦ ❡✱ ♣♦rt❛♥t♦✱ ✭✐✮✭❛✮ ♦❝♦rr❡✳ G

  ❙❡ ∗ 6= Id| ✱ ♥♦t❡ q✉❡ ♦ ▲❡♠❛ ✷✳✶✸ ❣❛r❛♥t❡ ✭✐✮✭❜✮ ❡✱ ♣♦rt❛♥t♦✱ ✭✐✮ s❡ ✈❡r✐✜❝❛✳ P❛r❛ ✈❡r✐✜❝❛r ♦s ✐t❡♥s ✭✐✐✮✲✭✐✈✮✱ ❜❛st❛ ❛♣❧✐❝❛r ♦s ✐t❡♥s ✭✐✐✐✮✱ ❞♦s ▲❡♠❛s ✷✳✼✱ ✷✳✽ ❡ ♦

  ▲❡♠❛ ✷✳✶✷✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ♦s ✐t❡♥s ✭✐✮✲✭✐✈✮ s❡❥❛♠ ✈❡r❞❛❞❡✐r♦s✳ P❛r❛ ♠♦str❛r q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ é s✉✜❝✐❡♥t❡ ✈❡r✐✜❝❛r q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ 1 2

  ❣❡r❛❞♦r❡s A = A ∪ A s❛t✐s❢❛③ ❡ss❛ ✐❞❡♥t✐❞❛❞❡✳ ❙❡❥❛♠ x, y /∈ G ❀ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ ∗ ∗ ∗ ∗

  (x )(y ) + (y )(x ) = 0, − σ(x)x − σ(y)y − σ(y)y − σ(x)x

  ♦✉ s❡❥❛ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ xy + yx + σ(xy)x y + σ(xy)y x x y ∗ ∗ ∗ ∗ ∗ ∗ − σ(y)xy − σ(y)y − σ(x)yx − σ(x)x = sx = sx y = xsy = xy x = ysx = yx s❡ ❛♥✉❧❛✳ ❈♦♠♦✱ ♣♦r ✭✐✮✱ x ❡ y ✱ ❡♥tã♦ x ❡ y ❀ ∗ ∗ ∗ ∗ 2 2 ∗ ∗

  ) = (sx) = x s = xs = 1 y ❛❧é♠ ❞✐ss♦ x = (x ✱ ♦✉ s❡❥❛✱ s ❡✱ ♣♦rt❛♥t♦✱ xy = x ✳ ❉❡ss❛ ❢♦r♠❛✱ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ s❡ r❡❡s❝r❡✈❡ ♣♦r ∗ ∗ (1 + σ(xy))(xy + yx) + y x). ∗ ∗ − (σ(x) + σ(y))(xy ✭✷✳✸✮

  = xys = ysx = y x ❙❡ (x, y) = 1✱ ❡♥tã♦ xy ❡ ❛ ❡①♣r❡ssã♦ s❡ s✐♠♣❧✐✜❝❛ ❡♠

  2(1 + σ(xy))xy , − 2(σ(x) + σ(y))xy 2

  ❡ ❡st❛ ❞❡✈❡ s❡r ♥✉❧❛ ♣❡❧♦ ✐t❡♠ ✭✐✐✮✱ ❥á q✉❡ (1 + σ(xy)) ❡ (σ(x) + σ(y)) ∈ R ❀ s❡ (x, y) 6= 1✱ ∗ ∗ x ♣♦r ✭✐✮✭❜✮✭✷✮✱ t❡♠♦s q✉❡ (x, y) = s✱ ❛ss✐♠ xy = yxs = y ❡ yx = xys = xy ✱ ♣♦rt❛♥t♦ ✭✷✳✸✮ ♣♦❞❡ s❡r r❡❡s❝r✐t❛ ❝♦♠♦

  ✷✽ ❡ ❡st❛ s❡ ❛♥✉❧❛✱ ❥á q✉❡ ♣❡❧♦ ✐t❡♠ ✭✐✐✮ t❡♠♦s 1 + σ(xy) = σ(x) + σ(y)✳ ∗ ∗

  ❙❡❥❛♠ ❛❣♦r❛ x /∈ G ❡ y ∈ G ✳ ❱❛♠♦s ♠♦str❛r q✉❡ s❡ α ∈ R é t❛❧ q✉❡ αy ∈ RG ✱ )

  ❡♥tã♦ αy ❛♥t✐❝♦♠✉t❛ ❝♦♠ (x − σ(x)x ✱ ♦✉ s❡❥❛ ∗ ∗ 0 = (x )αy + αy(x ) − σ(x)x − σ(x)x ∗ ∗ = α(xy + yx y ).

  − σ(x)x − σ(x)yx ∗ ∗ y = yx ❙❡ (x, y) = 1✱ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ q✉❡ ✜③❡♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ x ❡ ♣♦rt❛♥t♦ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ s❡ t♦r♥❛

  2α(xy y), − σ(x)x 2

  ♦ q✉❡ ❞❡ ❢❛t♦ s❡ ❛♥✉❧❛✱ ❥á q✉❡ ♣♦r ✭✐✐✐✮✱ α ∈ R ❀ s❡ (x, y) 6= 1✱ ♥♦✈❛♠❡♥t❡✱ ♣♦❞❡♠♦s ♠♦str❛r ∗ ∗ y q✉❡ xy = yx ❡ yx = x ✱ ♣♦rt❛♥t♦ t❡♠♦s q✉❡

  α(1 − σ(x))(xy + yx),

  ♥♦✈❛♠❡♥t❡ s❡ ❛♥✉❧❛ ♣♦r ✭✐✐✐✮✱ ❥á q✉❡ α(1 − σ(x)) = 0✳ P♦r ✜♠✱ ❛❞♠✐t✐♥❞♦ q✉❡ x, y ∈ G ✱ ❞❡✈❡♠♦s ✈❡r✐✜❝❛r q✉❡ s❡ α, β sã♦ t❛✐s q✉❡

  αx, βy ∈ RG ✱ ❡♥tã♦ ❡st❡s ú❧t✐♠♦s ❡❧❡♠❡♥t♦s ❛♥t✐❝♦♠✉t❛♠ ❡♥tr❡ s✐✱ ♦✉ s❡❥❛ 0 = αxβy + βyαx = αβ(xy + yx).

  ❙❡ (x, y) 6= 1✱ ♣♦r ✭✐✈✮ s❛❜❡♠♦s q✉❡ αβ = 0 ❡ ♦ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦❀ s❡ (x, y) = 1✱ ❡♥tã♦ 2 αβ

  ∈ R ❡ ♥♦✈❛♠❡♥t❡ t❡♠♦s ♦ r❡s✉❧t❛❞♦✳ ❈♦♠ ✐ss♦✱ ❝♦♥❝❧✉í♠♦s ❛ r❡❝í♣r♦❝❛✳ ❖ s❡❣✉✐♥t❡ ❝♦r♦❧ár✐♦ r❡ss❛❧t❛ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ R✱ G ❡ ∗ q✉❡✱ ❡♠❜♦r❛

  ♥ã♦ s❡❥❛♠ ♥❡❝❡ssár✐❛s ♣❛r❛ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ RG ✱ ❡st❛s sã♦ r❡❧❡✈❛♥t❡s ❡ ♠❡r❡❝❡♠ ❞❡st❛q✉❡✳ ❈♦r♦❧ár✐♦ ✷✳✶✺✳ ❙❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦✿

  ✭✐✮ σ(x) 6= −1, ∀x ∈ G ❀ ✭✐✐✮ ❙❡ ∗ = Id✱ ❡♥tã♦ exp(G/N) = 2❀ s❡ ∗ 6= Id✱ ❡♥tã♦ charR = 4 ❡ exp(G/N) = 2 ♦✉

  4 ❀

  ✭✐✐✐✮ ❙❡ ∗ 6= Id ❡ exp(G/N) = 2✱ ❡♥tã♦ s ∈ N✳ ) = ✭✐✈✮ ❙❡ (G {1}✱ ❡♥tã♦ G é ❛❜❡❧✐❛♥♦ ♦✉ ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ∗✳

  ❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ q✉❡✱ ✭✐✮ é ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ▲❡♠❛ ✷✳✶✱ ♦ ▲❡♠❛ ✷✳✷ ❣❛r❛♥t❡ ❛ ❝❛✲ r❛❝t❡ríst✐❝❛ ❞❡ R ❡♠ ✭✐✐✮✱ ❡ ❛ Pr♦♣♦s✐çã♦ ✷✳✶✵ ✐♠♣❧✐❝❛ ❡♠ ✭✐✈✮✳ P❛r❛ ✈❡r✐✜❝❛r exp(G/N) ❡♠ ✭✐✐✮ ♥♦t❡ q✉❡✱ s❡ ∗ = Id✱ ❡♥tã♦ ♣❡❧❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡

   2 ✷✾ q✉❡ G 6= N❀ s❡ ∗ 6= Id✱ t♦♠❛♥❞♦ x /∈ G ✱ s❛❜❡♠♦s✱ ♣❡❧♦ ▲❡♠❛ ✷✳✷✱ q✉❡ x ∈ G ❡ ♣❡❧♦ 4 ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❛♥t❡r✐♦r✱ ✈❡r✐✜❝❛♠♦s q✉❡ x ∈ N✱ ♦✉ s❡❥❛ exp(G/N) é ✉♠ ❞✐✈✐s♦r ❞❡ 4 ❡ ❞✐❢❡r❡♥t❡ ❞❡ 1✱ r❡st❛♥❞♦ ❝♦♠♦ ♣♦ss✐❜✐❧✐❞❛❞❡ 2 ♦✉ 4✳ 2 P♦r ✜♠✱ ♣❛r❛ ✈❡r✐✜❝❛r ✭✐✐✐✮✱ ♥♦t❡ q✉❡ s❡ exp(G/N) = 2✱ ❡♥tã♦ x −1

  ∈ N✱ ♦✉ s❡❥❛ σ(x) = σ(x)

  ) = 1 ✳ ❉❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❛ ✐♥✈♦❧✉çã♦ ❝♦♠ ❛ ♦r✐❡♥t❛çã♦✱ t❡♠♦s σ(xx ✱ ∗ x ) = σ(x) σ(x ) = σ(x)σ(x ) = 1 −1 ∗ −1 ∗ ∗

  ❛ss✐♠✱ s❡ x /∈ G ✱ ❡♥tã♦ σ(s) = σ(x ✳ ❱❛❧❡ ❧❡♠❜r❛r q✉❡ ♥♦ t❡♦r❡♠❛ ❛❝✐♠❛✱ ✉t✐❧✐③❛♠♦s ❛♣❡♥❛s ❛ ❡str✉t✉r❛ ❞❡ N ❞❡s✲

  ❝r✐t❛ ♣♦r ❬●P✶✸❛❪✱ ❡♠❜♦r❛ s❡❥❛♠ ❝♦♥❤❡❝✐❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ s✉❜❣r✉♣♦ C ≤ G✱ ❝♦♠ C =

  {x ∈ G : σ(x) = ±1}✱ ❡♥❝♦♥tr❛❞❛s ❡♠ ❬●P✶✸❜❪✳ ❙❡♥❞♦ ❛ss✐♠✱ ❝♦♠♦ ❛ ♣r♦✈❛ ❞❡st❡ r❡s✉❧t❛❞♦ ✐♥❞❡♣❡♥❞❡ ❞♦ ❝❛s♦ ❡♠ q✉❡ ❛ ♦r✐❡♥t❛çã♦ é ✉s✉❛❧✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ❡ss❡ r❡s✉❧t❛❞♦ ♠❛✐s ❣❡r❛❧ ❡ ❡♥❝♦♥tr❛r ❛ ♠❡s♠❛ ❞❡s❝r✐çã♦ ❝♦♥t✐❞❛ ♥❛ ❧✐t❡r❛t✉r❛✳ ❈♦r♦❧ár✐♦ ✷✳✶✻ ✭❚❡♦r❡♠❛ ✷✳✶✱ ❬●P✶✸❜❪✮✳ ❙❡❥❛ σ : G → {±1} ✉♠❛ ♦r✐❡♥t❛çã♦ ♥ã♦ tr✐✲ ✈✐❛❧ ❡♠ G✱ ∗ ✉♠❛ ✐♥✈♦❧✉çã♦ ❝♦♠♣❛tí✈❡❧ ❝♦♠ σ ❡ R ✉♠ ❛♥❡❧ ❝♦♠ charR 6= 2✳ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ charR = 4 ❡ ♦❝♦rr❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿ N = Id N = xs,

  ✭✶✮ G é ❛❜❡❧✐❛♥♦✱ ∗ | ❡ ❡①✐st❡ s ∈ N t❛❧ q✉❡ x ∀x / ∈ N✳ = xs,

  ✭✷✮ G é ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ∗ ❛ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ❡ x ∀x / ∈ N✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ P❡❧♦ ✐t❡♠ ✭✐✮ ❞♦ ❈♦r♦❧ár✐♦ ✷✳✶✺✱ t❡♠♦s q✉❡ G \N = ∅✱ ♣♦rt❛♥t♦ ∗ 6= Id ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣❡❧♦ ✐t❡♠ ✭✐✐✮✱ q✉❡ charR = 4✳ ❆♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✷✳✶✹✱ ❡♥❝♦♥tr❛♠♦s q✉❡ G é ❛❜❡❧✐❛♥♦ ♦✉ ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r

  = sx, ∗ ∗ = sx, ♥ã♦ tr✐✈✐❛❧ s ❡ x ∀x / ∈ G ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❝♦♠♦ G \N = ∅✱ t❡♠♦s x ∀x / ∈ N✳

  P❡❧♦ ❚❡♦r❡♠❛ ✷✳✻ ✭♦✉ ♦ ❚❡♦r❡♠❛ ✷✳✶✹ ♣❛r❛ σ ≡ 1✮ t❡♠♦s três ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ N

  ✱ N ✭❆✮ ∗ = Id| ❡ N é ❛❜❡❧✐❛♥♦❀ ✭❇✮ ∗ 6= Id✱ N é ❛❜❡❧✐❛♥♦ ❡ ❡①✐st❡ s ∈ N ∩ Z(G) t❛❧ q✉❡ x ∈ {x, sx} , ∀x ∈ N❀ ′ ∗

  = ✭❈✮ N {1, s} ❡ x ∈ {x, sx} , ∀x ∈ N✳

  ❙✉♣♦♥❤❛ q✉❡ G é ❛❜❡❧✐❛♥♦✳ ◆❡ss❡ ❝❛s♦✱ s❛❜❡♠♦s q✉❡ ✭❈✮ ♥ã♦ ♦❝♦rr❡ ❥á q✉❡ ❡st❡ ✐t❡♠ ✐♠♣❧✐❝❛ ❡♠ N ♣♦ss✉✐r ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧✳ ❱❛♠♦s ♠♦str❛r q✉❡ ✭❇✮ ♥ã♦ ♦❝♦rr❡ ❡✱ ❛ss✐♠ r❡st❛ ❝♦♠♦ ♣♦ss✐❜✐❧✐❞❛❞❡ ✭❆✮✱ ❡ ♦ ✐t❡♠ ✭✶✮ ❡stá ♣r♦✈❛❞♦✳

  = sx ❙❡ ✭❇✮ ♦❝♦rr❡✱ ❡♥tã♦ ❡①✐st❡ x ∈ N t❛❧ q✉❡ x ✳ ❚♦♠❛♥❞♦ y /∈ N✱ s❛❜❡♠♦s = sy ∗ ∗ q✉❡ y ✱ ❛❧é♠ ❞✐ss♦✱ xy /∈ N✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ xy /∈ G ✱ ♣♦✐s G \N = ∅✳ ❈♦♠♦ ∗ ∗ ∗ x, y, xy /

  = y x = yssx = yx ∈ G ✱ ❡♥tã♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✹✱ xy 6= (xy) ❡✱ ♣♦rt❛♥t♦✱ G é

  ♥ã♦ ❛❜❡❧✐❛♥♦✳ ❆ss✐♠ ❝♦♥❝❧✉í♠♦s ✭✶✮ ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ G é ♥ã♦ ❛❜❡❧✐❛♥♦✳ P❡❧♦ ✐t❡♠ ✭✐✈✮ ❞♦ ❈♦r♦❧ár✐♦ ✷✳✶✺ ❡ ♦ ❢❛t♦ ❞❡

  ✸✵ ❖❜s❡r✈❡ q✉❡ ❡ss❛ ❝♦♥❞✐çã♦ é tr✐✈✐❛❧♠❡♥t❡ s❛t✐s❢❡✐t❛ s❡ N é ❛❜❡❧✐❛♥♦✱ ♦✉ s❡❥❛✱ s❡ ✭❆✮ ♦✉ ✭❇✮ ♦❝♦rr❡✳ ❙✉♣♦♥❤❛ ❡♥tã♦ q✉❡ ❡st❛♠♦s ♥♦ ❝❛s♦ ✭❈✮ ❡ s❡❥❛♠ x ∈ G ⊂ N ❡ y / ∈ N✱ ∗ ∗

  = sy = sxy ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ xy /∈ N✱ ❡ ♥♦t❡ q✉❡✱ ❝♦♠♦ G ∗ ❡ (xy) ✱ ∗ ∗ ∗ ⊂ N✱ ❡♥tã♦ y

  = y x = syx ❛ss✐♠✱ sxy = (xy) ✱ ♦✉ s❡❥❛ (x, y) = 1❀ t♦♠❛♥❞♦ n ∈ N✱ t❡♠♦s q✉❡ ny /

  ∈ N ❡✱ ♣♦rt❛♥t♦✱ (x, ny) = 1✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ (x, n) = 1✱ ❥á q✉❡ (x, y) = 1✱ ♦✉ s❡❥❛✱ (G ∗ , G ∗ ) ∗ , N ) =

  ⊂ (G {1}✱ ❡ ❛ss✐♠✱ G é ✉♠ ❙▲❈✲❣r✉♣♦✱ ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s ✭✷✮✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ♥♦t❡ q✉❡ ♦s ✐t❡♥s ✭✐✮ ❡ ✭✐✐✮ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✹ sã♦ ❢❛❝✐❧♠❡♥t❡ ✈❡✲ ∗ ∗ r✐✜❝❛❞♦s✳ ❆❧é♠ ❞✐ss♦✱ t❛♥t♦ ✭✶✮ ❝♦♠♦ ✭✷✮ ✐♠♣❧✐❝❛♠ ❡♠ G ⊂ Z(G) ∩ N✱ ❧♦❣♦ s❡ x ∈ G ✱

  ♣❛r❛ q✉❡ αx ∈ RG ✱ ❞❡✈❡♠♦s t❡r α ·2 = α(1+σ(x)) = 0✱ ❧♦❣♦ ♦s ✐t❡♥s ✭✐✐✐✮ ❡ ✭✐✈✮ t❛♠❜é♠ s❡ ✈❡r✐✜❝❛♠✳ ❆ s❡❣✉✐r✱ ✈❛♠♦s ♠♦str❛r q✉❡ s❡ N s❛t✐s❢❛③ ♦ ✐t❡♠ ✭❆✮ ❞♦ ❚❡♦r❡♠❛ ✷✳✻ ❡ G ⊂ N✱

  ❡♥tã♦ ♣♦❞❡♠♦s ❞❡s❝r❡✈❡r ♦ q✉♦❝✐❡♥t❡ G/N ❡ ❡①✐❜✐r ❝♦♠ ♠❛✐s ❞❡t❛❧❤❡s ❛ ❡str✉t✉r❛ ❞❡ G✳ = N ◆♦t❡ q✉❡ ❡st❛s ❞✉❛s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s ❛ G ✳ Pr♦♣♦s✐çã♦ ✷✳✶✼✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦✱ G ✉♠ ❣r✉♣♦ ❝♦♠ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❡ σ : G

  → U(R) ✉♠❛ ♦r✐❡♥t❛çã♦ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦ t❛❧ q✉❡ exp(G/N) = 2✱ ❝♦♠ N = kerσ = N

  ❡ G ∗ ✱ ❡♥tã♦ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ charR = 4✱ ❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s✿

  ✭✶✮ G é ❛❜❡❧✐❛♥♦✱ ❡①✐st❡ s ∈ N ∩ Z(G) t❛❧ q✉❡ x ∈ {x, sx} ❡ σ é ✉♠❛ ♦r✐❡♥t❛çã♦ ❝❧áss✐❝❛✳ 2 2

  ✭✷✮ ✭❛✮ G é ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ∗ ❛ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ❡ G/N ≃ C × C ✳ ✭❜✮ ∀x, y /∈ G ✱ s❡ xy 6= yx✱ ❡♥tã♦ 1 + σ(xy) = σ(x) + σ(y)❀ s❡ xy = yx✱ ❡♥tã♦

  (1 + σ(xy)) 2 2 , ❡ (σ(x) + σ(y)) ∈ R ✳ ✭❝✮ α ∈ R ∀α ∈ R t❛❧ αx ∈ RG ✳ = N

  ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ RG s❡❥❛ ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❈♦♠♦ G 6= G✱ ❡♥tã♦ ♦ ∗ ∗ = N ❈♦r♦❧ár✐♦ ✷✳✶✺ ❣❛r❛♥t❡ q✉❡ charR = 4✳ ❈♦♠♦ G ✱ ❡♥tã♦ ❞❛❞♦s x, y ∈ G t❡♠♦s ∗ ∗ ∗ xy = x y = (yx) = yx ) = ∗ ∗

  ✱ ❥á q✉❡ yx ∈ N = G ✱ ♦✉ s❡❥❛ (G {1}✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ♣❡❧♦ ✐t❡♠ ✭✐✈✮ ❞♦ ❈♦r♦❧ár✐♦ ✷✳✶✺ q✉❡ G é ❛❜❡❧✐❛♥♦ ♦✉ ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ∗✳

  ❙✉♣♦♥❤❛ q✉❡ G s❡❥❛ ❛❜❡❧✐❛♥♦✳ ∗ = N ❙❡❥❛♠ x, y /∈ G ✳ P❡❧♦ ✐t❡♠ ✭❛✮ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✹ s❛❜❡♠♦s q✉❡ s❡ xy = yx = ∗ ∗ ∗ −1 yssx = y x = (xy) ∗ = N N = yN

  ✱ ♦✉ s❡❥❛ xy ∈ G ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ xN = y ✱ ♦♥❞❡ ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❛ ❤✐♣ót❡s❡ exp(G/N) = 2✳ ❈♦♠♦ G é ❛❜❡❧✐❛♥♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✱ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ q✉❡ ♥ã♦ ♣❡rt❡♥ç❛ ❛ N ❡stá ♥❛ ♠❡s♠❛ ❝❧❛ss❡ ❧❛t❡r❛❧ xN ♣❛r❛ ✉♠ x /

  ∈ N ✜①♦✱ ♣♦rt❛♥t♦ N ♣♦ss✉✐ ❛♣❡♥❛s ❞✉❛s ❝❧❛ss❡s ❧❛t❡r❛✐s ❡♠ G✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ G/N 2

  ✸✶ ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡✱ G ♥ã♦ é ❛❜❡❧✐❛♥♦✱ ♦✉ s❡❥❛✱ G é ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ∗ ❛ =

  ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛✳ ◆❡st❡ ❝❛s♦✱ s❛❜❡♠♦s q✉❡ N = G Z(G)✱ ❧♦❣♦ ♦ ❚❡♦r❡♠❛ ✷✳✾ 2 2 ❣❛r❛♥t❡ q✉❡ G/N = G/Z(G) ≃ C ✱ ❣❛r❛♥t✐♥❞♦ ✭❛✮✳

  × C P❛r❛ ✈❡r✐✜❝❛r ✭❜✮✱ ❜❛st❛ ❛♣❧✐❝❛r ♦ ✐t❡♠ ✭✐✐✮ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✹✳ ◆♦t❡ q✉❡ s❡ αx ∈ RG

  ✱ ❝♦♠ α 6= 0✱ ❡♥tã♦ x ∈ G ∗ ⊂ Z(G)✱ ♣♦rt❛♥t♦✱ ❛♣❧✐❝❛♥❞♦ ♦ ✐t❡♠ ✭✐✐✐✮ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✹✱ ✈❡r✐✜❝❛♠♦s ✭❝✮✳

  ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ✭✶✮ s❡❥❛ s❛t✐s❢❡✐t♦✳ ◆❡st❡ ❝❛s♦✱ ♦ ❈♦r♦❧ár✐♦ ✷✳✶✻ ❣❛r❛♥t❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✳

  P♦r ✜♠✱ s✉♣♦♥❤❛ q✉❡ ✭✷✮ ♦❝♦rr❛✳ ❱❛♠♦s ♠♦str❛r q✉❡ ♦s ✐t❡♥s ✭✐✮✲✭✐✈✮ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✹ sã♦ s❛t✐s❢❡✐t♦s✳

  ❖ ✐t❡♠ ✭✐✮ ♥❛t✉r❛❧♠❡♥t❡ é ✈❡r✐✜❝❛❞♦ ♣♦r ❙▲❈✲❣r✉♣♦s ❡ ♦ ✐t❡♠ ✭✐✐✮ é ❡①❛t❛♠❡♥t❡ ♦ ✐t❡♠ ✭❜✮✳ =

  ❖❜s❡r✈❡ q✉❡ s❡ G Z(G)✱ ❡ αx ∈ RG ✱ t❡♠♦s xy = yx✱ ∀y ∈ G✳ P♦rt❛♥t♦ ❛ ❤✐♣ót❡s❡ ✭❝✮ ❣❛r❛♥t❡ ✭✐✐✐✮ ❡ ✭✐✈✮✳ ▲♦❣♦ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✷✳✶✹✱ ❡♥❝♦♥tr❛♠♦s q✉❡ RG é ❛♥t✐❝♦♠✉t❛t✐✈♦✳

  ❈♦♠ ❡ss❡ ú❧t✐♠♦ r❡s✉❧t❛❞♦✱ é ♣♦ssí✈❡❧ ❝♦♥str✉✐r ❡①❡♠♣❧♦s ❞❛s ❡str✉t✉r❛s tr❛❜❛✲ ❧❤❛❞❛s ❞❡ ❢♦r♠❛ s✐♠♣❧❡s✳ 8 4 4 8

  = ❊①❡♠♣❧♦ ✷✳✶✽✳ ❙❡❥❛ G = Q hx, yi ❡ R = Z × Z ✳ ❱✐♠♦s ♥♦ ❊①❡♠♣❧♦ ✶✳✽ q✉❡ Q é ✉♠ ❙▲❈✲❣r✉♣♦✱ ❝✉❥❛ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ é ❞❛❞❛ ♣♦r ( 2

  g, s❡ g ∈ {1, x } g = 2 x g, 8 4 4 ) ❝❛s♦ ❝♦♥trár✐♦. 4 4 ) =

  ❚♦♠❛♥❞♦ σ : Q → U(Z × Z ✱ ❡♠ q✉❡ U(Z × Z {(1, 1), (1, 3), (3, 1), (3, 3)}✱ ❞❛❞❛ 2 = ♣♦r σ(x) = (3, 1) ❡ σ(y) = (1, 3)✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ gg ∈ N = Z(G) = G {1, x }✱ 4 4 )Q 8

  ♣♦rt❛♥t♦ σ∗ é ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛ ❡♠ (Z × Z ✳ 2 2 = y 2 2

  ❖❜s❡r✈❡ q✉❡ G/N = {N, xN, yN, xyN} ❥á q✉❡ x ✱ ❞♦♥❞❡ G/N ≃ C × C ✳ ❈♦♠ ❡ss❛s ✐♥❢♦r♠❛çõ❡s ♥ã♦ é ❞✐❢í❝✐❧ ♠♦str❛r q✉❡ ♦ ✐t❡♠ ✭✷✮ ❞❛ Pr♦♣♦s✐çã♦ ✷✳✶✼ é s❛t✐s❢❡✐t♦✱ 4 4 8

  )Q ) σ ❧♦❣♦ ((Z × Z ∗ é ❛♥t✐❝♦♠✉t❛t✐✈♦✳

  ❈❛♣ít✉❧♦ ✸ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❙✐♠étr✐❝♦s

  ◆❡st❡ ❝❛♣ít✉❧♦ ✐r❡♠♦s ❡st✉❞❛r ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❡❧❡♠❡♥t♦s s✐♠étr✐❝♦s ❡♠ r❡❧❛çã♦ ❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛ σ∗✳ σ ∗ = RG + ▼♦str❛♠♦s ♥♦ ❈❛♣ít✉❧♦ ✶ q✉❡ (RG) é ❣❡r❛❞♦ ♣♦r 1 =

  S {αx : x ∈ G ❡ α(1 − σ(x)) = 0} 2 = : x S {x + σ(x)x ∈ G\G } . + + , 2 ❖❜s❡r✈❡ q✉❡ s❡ x ∈ N ✱ ❡♥tã♦✱ rx ∈ RG ❛♥t✐❝♦♠✉t❛✱ 2 2 2 2 2 ∀r ∈ R✱ ❛ss✐♠✱ s❡ RG x = x x = 0

  ❡♥tã♦ r −r ✱ ♦✉ s❡❥❛✱ 2r ✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ t♦♠❛♥❞♦ r = 1 ✭♦✉ ❛❧❣✉♠❛ ✉♥✐❞❛❞❡ q✉❛❧q✉❡r✮✱ ❡♥tã♦ charR = 2✳ ❈♦♠♦ ❥á ❝♦♠❡♥t❛♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ s❡ charR = 2✱ ❡♥tã♦ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ sã♦ ❝♦♥❝❡✐t♦s ✐❞ê♥t✐❝♦s✱ ❛ss✐♠ ❝♦♠♦ ♦ sã♦ + ♦s ❝♦♥❥✉♥t♦s RG ❡ RG ✱ ❡ ❡st❡ ♥ã♦ é ♦ ❝❛s♦ q✉❡ ♣r❡t❡♥❞❡♠♦s ❡st✉❞❛r ♥❡st❛ t❡s❡✳ ❉❡ss❛ + ❢♦r♠❛✱ ✐r❡♠♦s ❡♥❢r❛q✉❡❝❡r ✉♠ ♣♦✉❝♦ ❛ ❤✐♣ót❡s❡ ❞❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡♠ RG ✱ ♣❛r❛ + ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡♠ S ⊂ RG ✱ s❡♥❞♦ S ♦ ♠❛✐♦r s✉❜❝♦♥❥✉♥t♦ ❞❡ RG t❛❧ q✉❡ ❛ + ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ♥ã♦ t❡♥❤❛ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ charR = 2✳ +

  , ❈♦♠♦ rx ∈ RG ∀x ∈ N ❡ r ∈ R✱ ♥ã♦ é ♥❡❝❡ssár✐♦ r❡t✐r❛r♠♦s ❝♦♠♣❧❡t❛♠❡♥t❡ +

  ♦s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ N ❞♦s ❣❡r❛❞♦r❡s ❞❡ RG ✱ ❞❡✈❡♠♦s ❛♣❡♥❛s t♦♠❛r ♦ ❝✉✐❞❛❞♦ ❡s❝♦❧❤❡r ❝✉✐❞❛❞♦s❛♠❡♥t❡ s❡✉s ❝♦❡✜❝✐❡♥t❡s✳ ❉❡✈✐❞♦ ❛♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✱ ✈❡r✐✜❝❛♠♦s q✉❡ 2

  = 0 1 1 s♦❜ ♥♦ss❛s ❤✐♣ót❡s❡s✱ 2r ✱ ♣♦rt❛♥t♦✱ s✉❜st✐t✉✐r❡♠♦s S ♣♦r S ✱ s❡♥❞♦ ❡st❡ ú❧t✐♠♦ ❞❛❞♦ ♣♦r 1 = rx : x , r R n o ∗ ∗ p 2 S ∈ N ∈ ∪ {αx : x ∈ G \N, α(1 − σ(x)) = 0} , 22 2 2

  = R = ♦♥❞❡ R {r ∈ R : 2r = 0} ❡ {r ∈ R : r ∈ R }✳ ❆ss✐♠✱ t❡♠♦s q✉❡ S é ❣❡r❛❞♦ 1 2

  ♣♦r S ∪ S ✱ ❡ s♦❜ ❡ss❡ ❝♦♥❥✉♥t♦ ✐r❡♠♦s ❛❞♠✐t✐r ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡✳ ■r❡♠♦s ✈❡r ♠❛✐s ❛❞✐❛♥t❡ q✉❡ ♦s r❡s✉❧t❛❞♦s ❛❝❡r❝❛ ❞❛s ✐♥✈♦❧✉çõ❡s ♥ã♦ ♦r✐❡♥t❛❞❛s

  ❡ ♦r✐❡♥t❛❞❛s ✉s✉❛✐s✱ ❛❜♦r❞❛❞♦s ❡♠ ❬●P✶✸❛✱ ●P✶✹❪✱ ♣♦❞❡♠ s❡r ❡①t❡♥❞✐❞♦s ♣❛r❛ ❝❛s♦s ✉♠ ♣♦✉❝♦ ♠❛✐s ❣❡r❛✐s✳

  ✸✸ ✈á❧✐❞♦ ♣❛r❛ q✉❛❧q✉❡r t✐♣♦ ❞❡ ♦r✐❡♥t❛çã♦ ❡♠ G✳ ▲❡♠❛ ✸✳✶✳ ❙❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ♦❝♦rr❡✿ 2

  √ 2 2 R , G ✭✐✮ ⊂ R ∗ = Id| ❡ G é ❛❜❡❧✐❛♥♦✳

  √ ∗ ∗ 2 R 2 = R 2 , xx = x x , ✭✐✐✮ charR = 4, ❡ x ∈ G ∀x ∈ G✳

  √ 2 , 1 , r 2 R ❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡ s❡ x ∈ G ∀x ∈ G✱ ❡♥tã♦ G é ❛❜❡❧✐❛♥♦✳ ❙❡❥❛♠ r ∈ ❡ 1 2 x x, r x

  ∈ N✱ ❛ss✐♠ r ∈ S✱ ❡ ♣♦rt❛♥t♦ ❡st❡s ❛♥t✐❝♦♠✉t❛♠ ❡♥tr❡ s✐✱ ♦✉ s❡❥❛✱ 1 2 2 1 r xr x = xr x, 1 2 2 −r 1 2 2 r x = 0 r q✉❡ ✐♠♣❧✐❝❛ ❡♠ 2r ✱ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ r ∈ R ✱ ❥á q✉❡ RG é ❧✐✈r❡♠❡♥t❡ 2

  √ 2 2 R ❣❡r❛❞♦ ♣♦r G✳ ❉❡ss❛ ❢♦r♠❛ ⊂ R ✱ ♣r♦✈❛♥❞♦ ✭✐✮✳ G

  P❛r❛ ♣r♦✈❛r ✭✐✐✮✱ s✉♣♦♥❤❛ q✉❡ ∗ 6= Id ❡ s❡❥❛ x /∈ G ✳ P♦r ❤✐♣ót❡s❡✱ x + σ(x)x ❛♥t✐❝♦♠✉t❛ ❝♦♥s✐❣♦ ♣ró♣r✐♦✱ ♦✉ s❡❥❛ 2 ∗ ∗ ∗ 2 2 2(x + σ(x)xx + σ(x)x x + σ(x) (x ) ) = 0. 2 ∗ ∗ ∗ 2

  , x x, (x ) ❈♦♠♦ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ❡q✉❛çã♦ é 0 ❡ 2 6= 0✱ ❡♥tã♦ ❞❡✈❡♠♦s t❡r q✉❡ x ∗ ∗ ∗ 2 2 2 2 ∈ {xx }✳

  = (x ) = (x ) ❈♦♠♦ x 6= x ✱ ❛ ú♥✐❝❛ ♣♦ss✐❜✐❧✐❞❛❞❡ é x ✱ ❡ ♣♦rt❛♥t♦ x ∗ ✳ P❡❧♦ ♠❡s♠♦ ∗ ∗

  ∈ G = x x

  = 0 ♠♦t✐✈♦✱ xx ❡✱ ♣❛r❛ q✉❡ ❛ ❡q✉❛çã♦ s❡ ♠❛♥t❡♥❤❛✱ é ♥❡❝❡ssár✐♦ q✉❡ 4σ(x)xx ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ 4 = 0✱ ❥á q✉❡ σ(x) ∈ U(R) ❡ RG é ❧✐✈r❡♠❡♥t❡ ❣❡r❛❞♦ ♣♦r G✳

  √ 2 R ∗ ∗ G , y + σ(y)y

  ❙❡❥❛ ❛❣♦r❛ r ∈ \ {0} ❡ y / ∈ G ✳ ❈♦♠♦ 1 ∈ N ✱ ❡♥tã♦ r = r1 ∈ S✱ ❧♦❣♦ ❛♥t✐❝♦♠✉t❛♠✱ ❛ss✐♠ ∗ ∗ r(y + σ(y)y ) = )r, −(y + σ(y)y

  ) = 0 ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ 2r(y + σ(y)y ❀ ♣❡❧♦ ❢❛t♦ ❞❡ y /∈ G ✱ ❝♦♥❝❧✉í♠♦s q✉❡ 2ry = 0✱ ❡✱

  √ √ 2 2 2 2 R = R R ♣♦rt❛♥t♦✱ 2r = 0✳ ❆ss✐♠ ❥á q✉❡ R ⊂ ✱ ❧♦❣♦✱ ♦ ✐t❡♠ ✭✐✐✮ s❡❣✉❡✳ G + ❈♦r♦❧ár✐♦ ✸✳✷✳ ❙❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦ ❡ ∗ 6= Id| ✱ ❡♥tã♦✱ ❞❛❞♦ x ∈ G ✱ s❡♠♣r❡ ❡①✐st❡ α

  ∈ R\ {0} t❛❧ q✉❡ αx ∈ RG ✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ σ(x) = −1✱ ❡♥tã♦ ❜❛st❛ t♦♠❛r α = 2✱ ❥á q✉❡ charR = 4❀ s❡ σ(x) 6= −11✱ 2

  ❡♥tã♦ α = 1 + σ(x) 6= 0 ❡ s❛t✐s❢❛③ ❛ ❡q✉❛çã♦✱ ♣♦✐s (1 + σ(x))(1 − σ(x)) = 1 − σ(x) ✱ ❡ 2 = 1

  ❝♦♠♦ x ∈ G ✱ ♣❡❧❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❛ ♦r✐❡♥t❛çã♦ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦✱ t❡♠♦s σ(x) ✳ ❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ é ❛♥á❧♦❣♦ ❛♦ ▲❡♠❛ ✷✳✹ ❡✱ ❛ss✐♠ ❝♦♠♦ ❡st❡✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ∗ ∗

  = y x q✉❡ xy = yx s❡✱ ❡ s♦♠❡♥t❡ s❡✱ xy ✱ s❡rá ✉s❛❞❛ s❡♠ ❢❛③❡r r❡❢❡rê♥❝✐❛ à ♠❡s♠❛✳ ▲❡♠❛ ✸✳✸✳ ❙✉♣♦♥❤❛ q✉❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❉❛❞♦s x, y ∈ G✱ ❡♥tã♦✱ xy = yx s❡✱ ❡ ∗ ∗ ∗ ∗ ∗ ∗ y = yx y = y x s♦♠❡♥t❡ s❡✱ x ✳ ❆❧é♠ ❞✐ss♦✱ s❡ xy = yx ❡ x, y /∈ G ✱ ❡♥tã♦ xy = yx = x ✱

  ✸✹ ❉❡♠♦♥str❛çã♦✳ P❛r❛ ♠♦str❛r♠♦s ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ G✱ ❜❛st❛ ♣r♦❝❡❞❡r ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ à ♣r♦✈❛ ❞♦ ▲❡♠❛ ✷✳✹✱ ♦❜s❡r✈❛♥❞♦ ❛♣❡♥❛s q✉❡ ❝❛s♦ x, y /∈ G ✱ ❡♥tã♦ ∗ ∗ (x + σ(x)x ) )

  ❛♥t✐❝♦♠✉t❛ ❝♦♠ (y + σ(y)y ❡✱ ♣♦rt❛♥t♦✱ ❞❡✈❡♠♦s s✉❜st✐t✉✐r ❛ ❡q✉❛çã♦ ✭✷✳✶✮✱ ♣♦r ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ xy + yx + σ(xy)x y + σ(xy)y x + σ(y)xy + σ(y)y x + σ(x)yx + σ(x)x y = 0. 2 ✭✸✳✶✮

  ❘❡st❛ ❛♣❡♥❛s ✈❡r✐✜❝❛r q✉❡ (1 + σ(xy)), (σ(x) + σ(y)) ∈ R ❝❛s♦ xy = yx ❡ x, y /∈ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ G y = y x = y x = x y = yx

  ✳ P❛r❛ ✐ss♦✱ ❝♦♠♦ xy = yx = x ❡ xy ✱ s✉❜st✐t✉✐♥❞♦ ♥❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r✱ ❡♥❝♦♥tr❛♠♦s

  2(1 + σ(xy))xy + 2(σ(x) + σ(y))x y = 0, ❡ ❝♦♠♦ x /∈ G ∗ ✱ ❢❛❝✐❧♠❡♥t❡ ❞❡❞✉③✐♠♦s ♦ q✉❡ ♣r♦❝✉r❛♠♦s✳

  ❙❡❣✉✐♥❞♦ ❛s ✐❞❡✐❛s ❛♣r❡s❡♥t❛❞❛s ♥♦ ❈❛♣ít✉❧♦ ✷✱ ✈❛♠♦s ❡st✉❞❛r ♦ ♣r♦❞✉t♦ ❡♥tr❡ ∗ ∗ ❡❧❡♠❡♥t♦s ❞♦s ❝♦♥❥✉♥t♦s G ❡ G\G ✳ ❱❡r❡♠♦s q✉❡ ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s s✐♠étr✐❝♦s é ✉♠ ❝❛s♦ ♠❛✐s ❝♦♠♣❧❡①♦ q✉❡ ♦ ❛♥t❡r✐♦r✱ ✈✐st♦ q✉❡ ♥♦ ❈❛♣ít✉❧♦ ✷✱ s❡ x /∈ G ✱ ❡♥tã♦ xy

  ∈ {yx, yx } ♣❛r❛ t♦❞♦ y ∈ G✱ ❛♦ ♣❛ss♦ q✉❡ ❛❣♦r❛ ♥ã♦ ❝♦♥s❡❣✉✐r❡♠♦s ✉♠❛ ✐♥❢♦r♠❛çã♦ ∗ ∗ ∗ ∗ , y x, x y tã♦ ♣r❡❝✐s❛✱ ❡ s✐♠ q✉❡ xy ∈ {yx, yx } ♣❛r❛ x, y / ∈ G ✳ ❱❡r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡ q✉❡

  ❡st❡ ❢❛t♦ ✐rá ❝✉❧♠✐♥❛r ♥✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❛ ♠❛✐s ♥❛ ❝❧❛ss✐✜❝❛çã♦ ❞❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s s✐♠étr✐❝♦s✳ y

  ▲❡♠❛ ✸✳✹✳ ❙✉♣♦♥❤❛ q✉❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❙❡ x, y /∈ G ❡ σ(y) 6= −1✱ ❡♥tã♦✱ x ∈ ∗ ∗ ∗ ∗ ∗ , x y = x y

  {x } ♦✉✱ σ(xy) = −1✱ xy = x ❡ xy ✳ ∗ ∗ ), (y + σ(y)y )

  ❉❡♠♦♥str❛çã♦✳ P♦r ❤✐♣ót❡s❡ t❡♠♦s (x + σ(x)x ∗ ∗ ∗ ∗ ∈ S✱ ❛ss✐♠ (x + σ(x)x )(y + σ(y)y ) = )(x + σ(x)x ),

  −(y + σ(y)y ❧♦❣♦ ✈❛❧❡ ❛ ❡q✉❛çã♦ ✭✸✳✶✮✳ y ∗ ∗

  / ❙✉♣♦♥❤❛ q✉❡ x ∈ {x, x } ♦✉ s❡❥❛✱ xy / ∈ {yx, yx }✳ ❈♦♠♦ x, y / ∈ G ✱ ♣❛r❛ q✉❡ ❛ ∗ ∗ ∗ ∗ ∗ x, y x , x y

  ❡q✉❛çã♦ ❛❝✐♠❛ s❡❥❛ ✈á❧✐❞❛✱ é ♥❡❝❡ssár✐♦ q✉❡ xy ∈ {y }✳ ❱❛♠♦s ❛♥❛❧✐s❛r ♦s ✸ ❝❛s♦s✳ x ❈❛s♦ ✶✿ xy = y ✳ ❈♦♠♦ σ(y) 6= −1✱ ❞❡✈❡♠♦s t❡r q✉❡ xy ❞❡✈❡ s❡r ✐❣✉❛❧ ❛ ♠❛✐s ∗ ∗ ∗ y x =

  ✉♠ ❡❧❡♠❡♥t♦ ❞❡ss❡ ú❧t✐♠♦ ❝♦♥❥✉♥t♦✳ ◆❡st❡ ❝❛s♦✱ xy = x ✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ y ∗ ∗ ∗ ∗ ∗ xy = y x x = x y

  ✱ ♦ q✉❡ ✐♠♣❧✐❝❛r✐❛ ❡♠ x ∈ G ✱ ❝♦♥tr❛❞✐çã♦✳ ❆ss✐♠✱ t♦♠❛♥❞♦ xy = y ✱ ∗ ∗ ∗ x = x y = yx ❛♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦ ❡♠ t♦❞♦s ♦s ♠❡♠❜r♦s✱ ♦❜t❡♠♦s y ✱ ❧♦❣♦✱ s✉❜st✐t✉✐♥❞♦ ♥❛ ❡q✉❛çã♦ ✭✸✳✶✮✱ t❡♠♦s ∗ ∗

  (1 + σ(xy) + σ(y))xy + (1 + σ(xy) + σ(x))yx + σ(y)xy + σ(x)yx = 0,

  ∗ ∗ ✸✺ = yx s❡♥❞♦ q✉❡ ♣❛r❛ ❛ ❡q✉❛çã♦ ❢❛③❡r s❡♥t✐❞♦✱ ❞❡✈❡♠♦s t❡r xy ❡✱ ♣♦rt❛♥t♦✱ σ(x) = −σ(y)❀

  ❛❧é♠ ❞✐ss♦✱ 1 + σ(xy) + σ(y) = 0 = 1 + σ(xy) + σ(x) = 1 + σ(xy) − σ(y),

  ❧♦❣♦✱ ❢❛③❡♥❞♦ ❛ ❞✐❢❡r❡♥ç❛ ❞♦ ♣r✐♠❡✐r♦ ♣❡❧♦ ú❧t✐♠♦ ♠❡♠❜r♦✱ ♦❜t❡♠♦s✱ 2σ(y) = 0✱ ✉♠ ❛❜s✉r❞♦✱ ❥á q✉❡ σ(y) ∈ U(R) ❡ charR 6= 2✳ ∗ ∗ ∗ ∗ ∗ ∗ x y x = xy = ∗ ∗ ❈❛s♦ ✷✿ xy = y ✳ ❖❜s❡r✈❡ q✉❡ xy 6= x ✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦ y x y

  ✱ ✐♠♣❧✐❝❛r✐❛ ❡♠ xy = yx✱ ❝♦♥tr❛❞✐çã♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❡❧❛ ❡q✉❛çã♦ ✭✸✳✶✮✱ t❡♠♦s σ(xy) = )

  −1 ❡✱ ♣♦rt❛♥t♦✱ 2xy ∈ S✱ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ 2xy ❛♥t✐❝♦♠✉t❛ ❝♦♠ (x + σ(x)x ✱ ♦✉ s❡❥❛ ∗ ∗ 2 2xyx + 2σ(x)xyx + 2x y + 2σ(x)xx y = 0. 2 ∗ ∗ y = xyx ❉❡s❞❡ q✉❡ xy 6= yx ❡ x /∈ G ∗ ✱ t❡♠♦s q✉❡ x ✱ ♦✉ s❡❥❛✱ xy = yx ✱ ❡ ♣♦rt❛♥t♦ ✉♠❛ y ∗

  / ❝♦♥tr❛❞✐çã♦✱ ❥á q✉❡ ♣♦r ❤✐♣ót❡s❡ x ∈ {x, x }✳ ∗ ∗ y

  ❈❛s♦ ✸✿ xy = x ✳ ∗ ∗ ∗ ∗ ∗ ∗ x , x y, yx , xy , y x ❈❛s♦ ✸✳❛✿ σ(xy) 6= −1✳ ◆❡st❛s ❝♦♥❞✐çõ❡s xy ∈ {yx, y }✱ ♦ q✉❡ s❡ r❡❞✉③ ❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ♣♦r ❤✐♣ót❡s❡ ♦✉ ❛ ❛❧❣✉♠ ❞♦s ❝❛s♦s ❛♥t❡r✐♦r❡s✳ ∗ ∗ ∗ x ∗ ∗ ❈❛s♦ ✸✳❜✿ σ(xy) = −1✳ ◆❡st❡ ❝❛s♦✱ s❛❜❡♠♦s q✉❡ yx = y ❡ ♣♦rt❛♥t♦ xy ∈ ∗ ∗ y, yx

  = yx {x }✱ ❥á q✉❡ xy 6= yx✳ ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ xy ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ∗ ∗ −1 x y = y x

  ) ✳ ❈♦♠♦ σ(xy) = −1✱ ❡♥tã♦ σ(x) = −σ(y ✱ ❡ ❥á q✉❡ σ(y) 6= −1✱ ❡♥tã♦ x /∈ N✱ 2 y)

  ♣♦rt❛♥t♦ σ(x 6= −1✳ ❖❜s❡r✈❡ q✉❡ ❛té ♦ ♠♦♠❡♥t♦ ♣r♦✈❛♠♦s q✉❡ s❡ a, b /∈ G ❡ σ(b) 6= −1✱ ❡♥tã♦ ∗ ∗ ∗ ab , a b

  ∈ {ba, ba }✱ ❥á q✉❡ ♦s ❞❡♠❛✐s ❝❛s♦s s❡ r❡❞✉③✐r❛♠ ❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ▼♦str❛r❡♠♦s 2 2 y / ∗ ∗ y) ❛❣♦r❛ q✉❡ x ∈ G ✱ ❛ss✐♠✱ ❝♦♠♦ x /∈ G ❡✱ ♣❡❧♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✱ σ(x 6= −1✱ x y ∗ 2

  ♣♦❞❡r❡♠♦s ❛♣❧✐❝❛r t❛❧ ❢❛t♦ ❛ ❡ss❡ ♣❛r ❞❡ ❡❧❡♠❡♥t♦s✱ ❡♥❝♦♥tr❛♥❞♦ q✉❡ x 2 ∗ ∗ 2 ∈ {x, x } ♦✉ x(x y) = x (x y) 22 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 y / ∗ y) = y x x = yxx = yx x = xy x = xx y y

  ❉❡ ❢❛t♦ x ∈ G ✱ ♣♦✐s (x x y ∗ ∗ ∗ 2 2 2 6= x ✱ y) = x (x y) ❛ss✐♠✱ ♣❡❧❛ ♦❜s❡r✈❛çã♦ ❛❝✐♠❛✱ s❛❜❡♠♦s q✉❡ x ∈ {x, x } ♦✉ x(x ✳ ❯♠❛ x y y 2 ∗ ∗ ∗ ∗ ∗ ∗ 2 2 3 2

  = x / y) = x (x y) y = x y (x ) = ✈❡③ q✉❡ x ∈ {x, x }✱ t❡♠♦s q✉❡ x(x ✱ ❛ss✐♠ x 2 2 2 2 2 xy(x ) = xyx y = yx 3 ✱ ❥á q✉❡✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✱ x ∈ G ❀ ❧♦❣♦ x ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 x y = xyxx = x y x x = x yxx = y xxx = x y x x = yx

  ✱ ❝♦♥❝❧✉í♠♦s q✉❡ xy = y ✱ ∗ ∗ = x y

  ✉♠ ❛❜s✉r❞♦✳ P♦rt❛♥t♦ ❞♦ ♣r✐♠❡✐r♦ ♣❛rá❣r❛❢♦ ❞❡st❡ ❝❛s♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ xy ✳ ▲❡♠❛ ✸✳✺✳ ❙✉♣♦♥❤❛ q✉❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❙❡ x, y /∈ G ❡ σ(y) 6= −1✱ ❡♥tã♦ ♦❝♦rr❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s✿ ∗ ∗ ∗ ∗ y = y x

  ✭✐✮ xy = yx = x ❡ 2(1 + σ(xy)) = 0 = 2(σ(x) + σ(y))✳ ∗ ∗ ∗ ∗ = y x = x y

  ✭✐✐✮ xy = yx ❡ 1 + σ(x) + σ(y) + σ(xy) = 0✳

  ∗ ∗ ∗ ∗ ✸✻ y = y x

  ✭✐✈✮ xy = yx 6= x ❡ σ(x) = −1✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ (x, y) = 1✱ ❡♥tã♦ ♣❡❧♦ ▲❡♠❛ ✸✳✸ t❡♠♦s ✭✐✮✳ ∗ ∗ ∗ y ∗ ∗ ∗ ∗ ∗ ❙✉♣♦♥❤❛ q✉❡ (x, y) 6= 1✳ ❆ss✐♠✱ ♣❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ xy = yx ♦✉ xy = x ✱ x y = xy = x y ❡ σ(xy) = −1✳ ❙❡ xy = yx ✱ ♣♦r ✉♠ ❛r❣✉♠❡♥t♦ s✐♠✐❧❛r ❛♦ ❝❛s♦ ∗ ∗ ∗ ∗ xy = y x

  = y x = x y ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ❡♥❝♦♥tr❛♠♦s q✉❡ xy = yx ❀ ❛ss✐♠✱ ❛♣❧✐❝❛♥❞♦ ❛

  ✐♥✈♦❧✉çã♦ ❡ s✉❜st✐t✉✐♥❞♦ ❡ss❛s ✐♥❢♦r♠❛çõ❡s ♥❛ ❡q✉❛çã♦ ✭✸✳✶✮✱ ♦❜t❡♠♦s ✭✐✐✮✳ ∗ ∗ ∗ ∗ ∗ y = x y ❙❡ xy 6= yx ✱ ❡♥tã♦ σ(xy) = −1✱ xy = x ❡ xy ✱ ❛ss✐♠✱ ❛♣❧✐❝❛♥❞♦ ❛ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ x = yx = y x y = y x x = yx

  ✐♥✈♦❧✉çã♦✱ y ❡ yx ✳ ❉❡ss❛ ❢♦r♠❛ xy = x 6= yx ✱ ❡ y 6= ∗ ∗ xy = x y ✱ ♣♦rt❛♥t♦✱ s✉❜st✐t✉✐♥❞♦ ❡ss❛s ✐♥❢♦r♠❛çõ❡s ♥❛ ❡q✉❛çã♦ ✭✸✳✶✮ ❡ ♦❜s❡r✈❛♥❞♦ q✉❡ (x, y )

  6= 1✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ σ(x) = −σ(y)✱ ❡ ✭✐✐✐✮ ♦❝♦rr❡✳ ∗ ∗ ∗ y ❋✐♥❛❧♠❡♥t❡ s✉♣♦♥❤❛ q✉❡ xy = yx ❡ xy 6= x ✳ ❆♣❧✐❝❛♥❞♦ ♦ q✉❡ ♣r♦✈❛♠♦s ❛❝✐♠❛ ❛ x ❡ y✱ s❛❜❡♠♦s q✉❡ ♦❝♦rr❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s✿ ∗ ∗ ∗ ∗ y = yx = xy = y x

  ✭❛✮ x ✳ ∗ ∗ ∗ ∗ y = yx = y x = xy ✭❜✮ x ∗ ∗ ∗ ∗ y = xy x

  ✭❝✮ x 6= yx = y ✳ ∗ ∗ ∗ ∗ y = yx x = xy ✭❞✮ x 6= y ✳

  ❖❜s❡r✈❡ q✉❡ ♦s ✐t❡♥s ✭❛✮✱ ✭❝✮ ❡ ✭❞✮ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❛♦s ✐t❡♥s ✭✐✮✱ ✭✐✐✐✮ ❡ ✭✐✈✮✱ ∗ ∗ y = yx x = r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❧♦❣♦ ♥ã♦ ♦❝♦rr❡♠✱ ♣♦rt❛♥t♦✱ x ❡✱ ❛♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦✱ y ∗ ∗ x y

  ✳ ❉❡ss❛ ❢♦r♠❛✱ s✉❜st✐t✉✐♥❞♦ ❡♠ ✭✸✳✶✮✱ ❡♥❝♦♥tr❛♠♦s σ(x) = −1✳ ▲❡♠❛ ✸✳✻✳ ❙✉♣♦♥❤❛ q✉❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❙❡ x, y /∈ G ∗ ✱ ❡♥tã♦ sã♦ ✈❡r❞❛❞❡✐r❛s✿ ∗ ∗ ∗ ∗

  , y x, x y ✭✐✮ xy ∈ {yx, yx

  }✳ ✭✐✐✮ xy = yx s❡✱ ❡ s♦♠❡♥t❡ s❡✱ xy ∈ G ✳ ∗ ∗ y = yx

  ✭✐✐✐✮ xy = yx s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x ✳✴✴ ❉❡♠♦♥str❛çã♦✳ P❛r❛ ♣r♦✈❛r ✭✐✮✱ s✉♣♦♥❤❛ q✉❡ (x, y) 6= 1✳ ❙❡ σ(y) 6= −1✱ ❡♥tã♦✱ ♣❡❧♦ ▲❡♠❛ ✸✳✹✱ t❡♠♦s ♦ r❡s✉❧t❛❞♦✳ ❙✉♣♦♥❤❛ q✉❡ σ(y) = −1✳ ∗ ∗ / ∗ ∗ ∗ ∗ ∗ ∗ ❙❡ σ(x) 6= −1✱ ❝♦♠♦ y /∈ G ✱ ❡♥tã♦✱ y ∈ G ✱ ♣♦rt❛♥t♦✱ ❛♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✸✳✹ ❛ y x = xy )x = (y ) x = yx

  ❡ x✱ ♦❜t❡♠♦s q✉❡ y ♦✉ (y ❡ ❛♠❜♦s ✐♠♣❧✐❝❛♠ ❡♠ ✭✐✮✳ ∗ ∗ ) )

  ❙❡ σ(x) = −1✱ ♣♦r ❤✐♣ót❡s❡✱ (x − x ❛♥t✐❝♦♠✉t❛ ❝♦♠ (y − y ✱ ❧♦❣♦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ xy + yx + x y + y x y x = 0, ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ − x − yx − y − xy y , y x , x y, yx , y x, xy y, xy ❛ss✐♠✱ xy ∈ {yx, x }✳ P♦r ❤✐♣ót❡s❡✱ xy / ∈ {yx, x }✳ ❙✉✲

  ✸✼ ❝♦♠♦ charR = 4✱ 2xy ♥ã♦ s❡ ❛♥✉❧❛ ♥❛ ❡q✉❛çã♦ ❡ ♣♦rt❛♥t♦ ❡ss❛ ♥ã♦ ♣♦❞❡ s❡r ♥✉❧❛✱ ❝♦♥tr❛✲ ❞✐çã♦✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥❝❧✉✐♠♦s ✭✐✮✳ ∗ ∗ ∗ ∗

  , y x, x y P❛r❛ ♣r♦✈❛r ✭✐✐✮✱ s✉♣♦♥❤❛ q✉❡ (x, y) 6= 1✱ ❛ss✐♠✱ ♣❡❧♦ ✐t❡♠ ✭✐✮✱ xy ∈ {yx ∗ ∗ ∗ ∗ ∗ ∗ }✱ x x = xy = x y

  ♣♦rt❛♥t♦✱ s❡ xy ∈ G ∗ ✱ ❡♥tã♦ xy = y ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ y ✱ ❥á q✉❡ s❡ ∗ ∗ ∗ ∗ y x = xy x, yx ∈ {y }✱ ❡♥tã♦ x ♦✉ y ∈ G ∗ ✱ ✉♠ ❛❜s✉r❞♦❀ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❛♣❧✐❝❛♥❞♦ ❛ ∗ ∗ ∗ ∗ x = x y

  ✐♥✈♦❧✉çã♦ ❡♠ y ✱ t❡♠♦s xy = yx✱ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ❛ ❤✐♣ót❡s❡ ❞❡ (x, y) 6= 1✱ ∗ ∗ ∗ ∗ x = x y ❧♦❣♦ xy /∈ G ✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ xy = yx✱ ❛♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦✱ ♦❜t❡♠♦s y ∗ ∗ ∗ ∗ ∗

  = y x y = yx ) ❡✱ ❛♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✸✳✸✱ xy ❡ x ❀ ❝♦♠♦ (x + σ(x)x ❛♥t✐❝♦♠✉t❛ ❝♦♠ (y + σ(y)y )

  ✱ ❡♥❝♦♥tr❛♠♦s ❛ ❡q✉❛çã♦ ∗ ∗ ∗ ∗ ∗ ∗ 2xy + 2σ(xy)y x + 2σ(y)xy + 2σ(x)x y = 0, ∗ ∗ y x =

  ❡ ❝♦♠♦ xy 6= xy 6= x ✱ ♣❛r❛ q✉❡ ❡ss❛ ❡q✉❛çã♦ ❢❛ç❛ s❡♥t✐❞♦✱ ❞❡✈❡♠♦s t❡r xy = y (xy)

  ✳ ∗ ∗ y P❛r❛ ✈❡r✐✜❝❛r ✭✐✐✐✮✱ s✉♣♦♥❤❛ q✉❡ xy = yx ❡ x ∗ ∗ ∗ ∗ ∗ 6= yx✱ ♥❡ss❡ ❝❛s♦✱ (x, y) 6= 1 6=

  (x , y) y x , xy ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ✱ ♣♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ ✐t❡♠ ✭✐✮ ❛ x ❡ y ❡ ❡♥❝♦♥tr❛r q✉❡ x ∈ {y }✳ y = y x x = xy = yx y = y x = xy y = xy

  ❙❡ x ✱ ❡♥tã♦ y ✱ ❧♦❣♦✱ x ❀ s❡ x ✱ ❡♥tã♦ ∗ ∗ ∗ ∗ ∗ ∗ y x = yx = xy y = xy = y x y = ∗ ∗ ∗ ✱ ❧♦❣♦✱ x ✳ ❆ss✐♠✱ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s✱ t❡♠♦s x ∗ ∗ ∗ ∗ y x = xy x = xy = yx y

  6= yx✱ ❡ ❛♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦✱ ♦❜t❡♠♦s y 6= x ✱ ♣♦rt❛♥t♦✱ s✉❜st✐t✉✐♥❞♦ ❡ss❛s ✐♥❢♦r♠❛çõ❡s ❡♠ ✭✸✳✶✮✱ ❡♥❝♦♥tr❛r❡♠♦s σ(xy) + σ(x) + σ(y) = 1 + σ(x) + σ(y) = 1 + σ(xy) = 0,

  ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ❥á q✉❡ charR 6= 2✳ P❛r❛ ❛ r❡❝í♣r♦❝❛✱ ❜❛st❛ ❛♣❧✐❝❛r ♦ ♠❡s♠♦ r❡s✉❧t❛❞♦ ❛ x ❡ y✳

  ❆ss✐♠ ❝♦♠♦ ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ❛ ❞❡s❝r✐çã♦ ❞❡ q✉❛♥❞♦ S é ❛♥t✐❝♦♠✉t❛t✐✈♦ ❝❛s♦ σ s❡❥❛ ✉♠❛ ♦r✐❡♥t❛çã♦ tr✐✈✐❛❧ s❡rá ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❛ s❡❣✉✐r✱ ♣♦ré♠✱ ❛♦ ❝♦♥trár✐♦ ❞♦ q✉❡ ❛❧❝❛♥ç❛♠♦s ♥❡st❡ ♠❡s♠♦ ❝❛♣ít✉❧♦✱ ❡ss❛ ❝❛r❛❝t❡r✐③❛çã♦ ♥ã♦ s❡rá s✉✜❝✐❡♥t❡ ❡ ♣r❡❝✐s❛r❡♠♦s t❛♠❜é♠ ❞❛ ❞❡s❝r✐çã♦ ❞❡ q✉❛♥❞♦ S é ❛♥t✐❝♦♠✉t❛t✐✈♦ ❝❛s♦ σ s❡❥❛ ✉♠❛ ♦r✐❡♥t❛çã♦ ❝❧áss✐❝❛✳ ❚❛✐s r❡s✉❧t❛❞♦s s❡ ❡♥❝♦♥tr❛♠ ❡♠ ❬●P✶✸❛✱ ●P✶✹❪✳ ◆❡st❡s ❛rt✐❣♦s✱ ♦s

  √ R 2 = 2R

  ❛✉t♦r❡s ✐♠♣❧✐❝✐t❛♠❡♥t❡ r❡str✐♥❣✐r❛♠ ♦ ❡st✉❞♦ ❛♦ ❝❛s♦ ❡♠ q✉❡ ✱ ❞❡✐①❛♥❞♦ ❞❡ ❧❛❞♦ ♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧ q✉❡ ❛♣r❡s❡♥t❛r❡♠♦s ❛q✉✐✳

  ❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ❞❡s❝r❡✈❡ ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ S q✉❛♥❞♦ σ ≡ 1✱ ❡ ❡st❡ é ✉♠❛ ❡①t❡♥sã♦ ❞♦ ❬❚❡♦r❡♠❛ ✸✳✷✱ ●P✶✸❛❪✳ ❚❡♦r❡♠❛ ✸✳✼✳ ❙❡❥❛ R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ t❛❧ q✉❡ charR 6= 2 ❡ G ✉♠ ❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ∗✳ ❊♥tã♦ S ✭❝♦♠ σ tr✐✈✐❛❧✮ é ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦❝♦rr❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

  √ 2 2

  ✸✽ √ 2 2 2

  R = R = 1

  ✭✷✮ charR = 4✱ ✱ G é ❛❜❡❧✐❛♥♦ ❡ ❡①✐st❡ s ∈ G t❛❧ q✉❡ s ❡ x ∈ {x, sx} , ∀x ∈ G✳

  √ 2 2 ′ ∗ R = R =

  ✭✸✮ charR = 4✱ ✱ G {1, s} ❡ x ∈ {x, sx} , ∀x ∈ G✳

  ❉❡♠♦♥str❛çã♦✳ P❛r❛ ♣r♦✈❛r ❡ss❡ r❡s✉❧t❛❞♦✱ é s✉✜❝✐❡♥t❡ s❡❣✉✐r ❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✷ ❞❡ ❬●P✶✸❛❪✱ ✉s❛♥❞♦ ♦ ▲❡♠❛ ✸✳✶ ♥♦ ❧✉❣❛r ❞♦ ▲❡♠❛ ✸✳✶ ❞❡ ❬●P✶✸❛❪ ❡ s✉❜st✐t✉✐♥❞♦ ❛ ❡q✉❛çã♦ ✭✸✳✶✮ ❞♦ ♠❡s♠♦ ❛rt✐❣♦✱ ♥♦s ❝❛s♦s ❞❡ g ♦✉ h ∈ G ♣❡❧❛s s❡❣✉✐♥t❡s✿ ∗ ∗ ∗ ∗ r 1 (gh + gh + hg + h

  g) = 0, ❝❛s♦ g ∈ G ❡ h /∈ G ❀ ❡ 1 2 r r (gh + hg) = 0, 1 22 , r R

  ❝❛s♦ g, h ∈ G ✱ s❡♥❞♦ r ∈ ✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ✜③❡♠♦s ♥♦ ❈á♣✐t✉❧♦ ✷✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ♦ t❡♦r❡♠❛ ❛❝✐♠❛

  ♣❛r❛ ❞❡s❝r❡✈❡r ❛ ❡str✉t✉r❛ ❞♦ s✉❜❣r✉♣♦ N ≤ G✱ ❡✱ ❛ ♣❛rt✐r ❞✐ss♦✱ ❡①tr❛✐r ✐♥❢♦r♠❛çõ❡s ❛ r❡s♣❡✐t♦ ❞❡ G✳ ❊♥tr❡t❛♥t♦✱ ❝♦♠♦ ❥á ❢♦✐ ❞✐t♦✱ ❝♦♥❤❡❝❡r ❛ ❡str✉t✉r❛ ❞❡ N ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛

  ❝❛r❛❝t❡r✐③❛r♠♦s ❝♦♠♣❧❡t❛♠❡♥t❡ q✉❛♥❞♦ ♦ ❝♦♥❥✉♥t♦ S✱ ♣❛r❛ σ ✉♠❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐✲ ③❛❞❛✱ é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ P❛r❛ ❝♦♠♣❧❡t❛r♠♦s t❛❧ t❛r❡❢❛✱ ♣r❡❝✐s❛r❡♠♦s ❞❡ ♠❛✐s ❛❧❣✉♠❛s ✐♥❢♦r♠❛çõ❡s ❛❝❡r❝❛ ❞♦ ❣r✉♣♦ G❀ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ♥❡❝❡ss✐t❛r❡♠♦s ❝♦♥❤❡❝❡r ♦ s✉❜❣r✉♣♦ C =

  {x ∈ G : σ(x) = ±1}✳ P❛r❛ ❞❡s❝r❡✈❡r C✱ ♣♦❞❡♠♦s ♣r♦❝❡❞❡r ❞❡ ♠❛♥❡✐r❛ s❡♠❡❧❤❛♥t❡ à q✉❡ ✜③❡♠♦s ♣❛r❛

  N ✱ ❡st✉❞❛♥❞♦ ❛❣♦r❛ ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ S ❝❛s♦ σ s❡❥❛ ❛ ♦r✐❡♥t❛çã♦ ✉s✉❛❧✳ P❛r❛ ✐ss♦✱

  ♣♦❞❡♠♦s✱ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ ❡♥❝♦♥tr❛r ✉♠❛ ❡①t❡♥sã♦ ❞❛ ❞❡s❝r✐çã♦ ❛♣r❡s❡♥t❛❞❛ ❡♠ ❬●P✶✹❪✱ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✸✳✼ ♥♦ ❧✉❣❛r ❞♦ ❚❡♦r❡♠❛ ✸✳✷ ❞❡ ❬●P✶✸❛❪ ❡ ❛❥✉st❛♥❞♦ ❛❧❣✉♠❛s ❡q✉❛çõ❡s ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛ q✉❡ ✜③❡♠♦s ❛♥t❡r✐♦r♠❡♥t❡✳ ❚❡♦r❡♠❛ ✸✳✽✳ ❙❡❥❛ R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ t❛❧ q✉❡ charR 6= 2 ❡ G ✉♠ ❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ∗ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ✉♠❛ ♦r✐❡♥t❛çã♦ σ : G → {±1}✳ ❊♥tã♦ S é ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦❝♦rr❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s✿

  √ 2 2 R ✭✶✮ ⊂ R ✱ G é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❡ ∗ = Id✳

  √ 2 2 R = R N = Id N ✭✷✮ charR = 4✱ ✱ G é ❛❜❡❧✐❛♥♦ ❡ ∗| ✳

  √ ′ ∗ R 2 = R 2 = ) =

  ✭✸✮ charR = 4✱ ✱ G {1, s} ❡ x ∈ {x, sx} , ∀x ∈ G✳ ❆❧é♠ ❞✐ss♦✱ (G 2 2 = {1} ♦✉ R {0}✳ ′ ∗

  = 2 ◆♦t❡ q✉❡ G {1, s} ❡ x ∈ {x, sx}✱ ✐♠♣❧✐❝❛♠ q✉❡ s ∈ Z(G) ∩ G ✳ ❉❡ ❢❛t♦✱ = 1 = ss = 1

  ❝♦♠♦ s ✱ s❡ s /∈ G ✱ ❡♥tã♦ s ✱ ❞♦♥❞❡ s = 1✱ ❝♦♥tr❛❞✐çã♦❀ ♣♦r ♦✉tr♦ ❧❛❞♦

  −1 −1 −1 ✸✾ s xs = s sx = 1 s❡ (x, s) 6= 1 ♣❛r❛ ❛❧❣✉♠ x ∈ G✱ ❡♥tã♦ x ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ x ✱ ❝♦♥tr❛❞✐çã♦✱ ♣♦✐s ♥❡st❡ ❝❛s♦ s ♣♦ss✉✐r✐❛ ♦r❞❡♠ 1✳

  ❈♦♥❤❡❝❡♥❞♦ ❡ss❡s r❡s✉❧t❛❞♦s✱ ♣♦❞❡♠♦s ♣r♦ss❡❣✉✐r ♥♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❛s ♣r♦♣r✐❡✲ ❞❛❞❡s ❞♦ ❣r✉♣♦ G✳ 2

  , y) = 1, ∗ ▲❡♠❛ ✸✳✾✳ ❙❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ (x ∀x, y / ∈ G ✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ x, y /∈ G t❛✐s q✉❡ (x, y) 6= 1✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣❡❧♦ ✐t❡♠ ✭✐✐✮ ❞♦

  = ▲❡♠❛ ✸✳✻✱ xy /∈ G ✳ ❙❡ x, y ∈ C✱ ♣❡❧♦ ✐t❡♠ ✭✸✮ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ G {1, s}✱ ❞♦♥❞❡ −1 −1 −1 −1 x x yxx = x syx = sx yx = ssy = y

  ❡ ♦ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦✳ ❙❡ x ∈ C ❡ y /∈ C✱ ❡♥tã♦✱ ❛♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✸✳✺ ❛ x ❡ y✱ t❡♠♦s q✉❡ ✭✐✐✮ ♦✉ ✭✐✈✮ ∗ ∗ ∗ 2 2 2 y = xyx = y(x ) = yx

  ♦❝♦rr❡✱ ❥á q✉❡ xy /∈ C✱ ❧♦❣♦✱ xy = yx ✳ ❆ss✐♠ x ✳ ❙❡ x /∈ C ❡ y ∈ C✱ ♣♦❞❡♠♦s ♣r♦❝❡❞❡r ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✳

  ❙❡ x, y /∈ C✱ ❛♣❧✐❝❛♥❞♦ ♦ ♠❡s♠♦ ❧❡♠❛✱ t❡♠♦s q✉❡ s❡ ✭✐✐✮ ♦❝♦rr❡✱ ♦ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦ ♣❡❧♦s ♠❡s♠♦s ❛r❣✉♠❡♥t♦s ❛❝✐♠❛❀ ❛❧é♠ ❞✐ss♦ ✭✐✮ ❡ ✭✐✈✮ ♥ã♦ sã♦ ✈á❧✐❞♦s✳ ❙✉♣♦♥❤❛ ❡♥tã♦ q✉❡ ∗ ∗ ∗ ∗ y = y x ✭✐✐✐✮ ♦❝♦rr❛✱ ♦✉ s❡❥❛✱ xy = x 6= yx ❡ σ(xy) = −1✳ ❆♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ ▲❡♠❛ 2 y /

  ✸✳✺ ❛ xy ❡ x✱ t❡♠♦s q✉❡ ✭✐✐✐✮ ♥ã♦ ♦❝♦rr❡✱ ❥á q✉❡ x ∈ C✳ ❙❡ ✭✐✮ ♦❝♦rr❡✱ ❡♥tã♦ xyx = xxy✱ x = (yx)x ♦✉ s❡❥❛✱ xy = yx✱ ❛❜s✉r❞♦✳ ❙❡ ✭✐✐✮ ♦✉ ✭✐✈✮ ♦❝♦rr❡✱ ❡♥tã♦ x(xy) = (xy) ✱ ♦✉ s❡❥❛✱ 2

  (x , y) = 1 ✳ G

  ❆♥t❡s ❞❡ ❛✈❛♥ç❛r♠♦s ♣❛r❛ ♦ ♣ró①✐♠♦ ❧❡♠❛✱ ❞❡✈❡♠♦s ♦❜s❡r✈❛r q✉❡ s❡ ∗ 6= Id| ✱ ❡♥tã♦ ♣❛r❛ ❝❛❞❛ x ∈ G ✱ ❡①✐st❡ ❛♦ ♠❡♥♦s ✉♠ α ∈ R\ {0} t❛❧ q✉❡ αx ∈ S✳ P❛r❛ ✐ss♦✱ ♥♦t❡ G 22

  = R ∗ q✉❡ s❡ ∗ 6= Id| ✱ ❡♥tã♦ ♦ ▲❡♠❛ ✸✳✶✱ ❛✜r♠❛ q✉❡ 2 ∈ R ✱ ❡ ♣♦rt❛♥t♦✱ s❡ x ∈ G ❡ σ(x) = −1✱ t♦♠❛♥❞♦ α = 2✱ ❛ ❛✜r♠❛çã♦ é ✈á❧✐❞❛✱ ❥á q✉❡ 2(1 − σ(x)) = 2 · 2 = 0❀ s❡ σ(x)

  6= −1✱ ❡♥tã♦✱ t♦♠❛♥❞♦ α = 1 + σ(x)✱ t❡♠♦s q✉❡ α 6= 0 ❡ α(1 − σ(x)) = (1 + σ(x))(1 − 2 σ(x)) = 1 = 0 2 − σ(x) ✱ ❥á q✉❡✱ ♣❡❧❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❛ ✐♥✈♦❧✉çã♦ ❝♦♠ ❛ ♦r✐❡♥t❛çã♦✱ x = xx

  ∈ N✳ ❖ ❧❡♠❛ ❛ s❡❣✉✐r ♠♦str❛ ❛s r❡❧❛çõ❡s ❡♥tr❡ ❡❧❡♠❡♥t♦s ❞❡ G ❝♦♠ ✉♠ ❡❧❡♠❡♥t♦s ❞❡

  G \G ✳ ∗ ∗

  ▲❡♠❛ ✸✳✶✵✳ ❙✉♣♦♥❤❛ q✉❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ y ∈ G ✱ x /∈ G ❡ α ∈ R t❛❧ q✉❡ αy ∈ S✱ sã♦ ✈❡r❞❛❞❡✐r❛s✿ y ∗ ✭✐✮ x

  ∈ {x, x }✳ ✭✐✐✮ xy ∈ G ⇔ xy 6= yx✳ 2

  ✭✐✐✐✮ ❙❡ xy 6= yx✱ ❡♥tã♦ α(1 + σ(x)) = 0✳ ❙❡ xy = yx✱ ❡♥tã♦ α ∈ R ✳ 2 2 ) = (x , y) = (xx , y) = 1

  ✭✐✈✮ (x, y ✳

  ✹✵ ❉❡♠♦♥str❛çã♦✳ ❖s ✐t❡♥s ✭✐✮✱ ✭✐✐✮ ❡ ✭✐✐✐✮ sã♦ ❛♥á❧♦❣♦s ❛♦s ♠❡s♠♦s ✐t❡♥s ❞♦ ▲❡♠❛ ✷✳✽✱ s❡♥❞♦ s✉✜❝✐❡♥t❡ s✉❜st✐t✉✐r ♦ ❈♦r♦❧ár✐♦ ✷✳✸ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✸✳✷✳

  P❛r❛ ✈❡r✐✜❝❛r ✭✐✈✮✱ ♥♦t❡ q✉❡ s❡ (x, y) 6= 1✱ ❡♥tã♦✱ ♣❡❧♦ ✐t❡♠ ✭✐✮✱ 2 ∗ ∗ 2 2 x y = xyx = y(x ) = yx , 2 2 xy = yx y = y x ∗ ∗ ∗ ❡ xx y = xyx = yx x = yxx .

  ❖ ♣ró①✐♠♦ ❧❡♠❛ ❞❡s❝r❡✈❡ ❛❧❣✉♠❛s r❡❧❛çõ❡s ❡♥tr❡ ❞♦✐s ❡❧❡♠❡♥t♦s s✐♠étr✐❝♦s ❡ s❡✉s ❝♦❡✜❝✐❡♥t❡s✳ ❆❧é♠ ❞✐ss♦✱ ♦ ✐t❡♠ ✭✐✈✮ ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r ❡ ♦ ✐t❡♠ ✭✐✐✮ ❞♦ s❡❣✉✐♥t❡✱ ❣❛r❛♥t❡♠ 2 q✉❡ x ∈ Z(G), ∀x ∈ G✳

  ▲❡♠❛ ✸✳✶✶✳ ❙✉♣♦♥❤❛ q✉❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ x, y ∈ G ❡ α, β ∈ R t❛✐s q✉❡ αx, βy ∈ S t❡♠♦s 2 ✭✐✮ ❙❡ xy 6= yx✱ ❡♥tã♦ αβ = 0❀ s❡ xy = yx✱ ❡♥tã♦ αβ ∈ R ✳ 2 2

  ) = (x , y) = 1 ✭✐✐✮ (x, y ✳

  ❉❡♠♦♥str❛çã♦✳ ❖ ✐t❡♠ ✭✐✮ é ❛♥á❧♦❣♦ ❛♦ ▲❡♠❛ ✷✳✶✷✳ P❛r❛ ♣r♦✈❛r ✭✐✐✮✱ s✉♣♦♥❤❛ q✉❡ (x, y) 6= 1✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ ♣❡❧❛ ❖❜s❡r✈❛çã♦ ✷✳✶✶✱ xy /

  = ∈ G ✳ ❙❡❥❛ α ♥ã♦ ♥✉❧♦ t❛❧ q✉❡ αx ∈ S✱ ❛ss✐♠✱ αx ❛♥t✐❝♦♠✉t❛ ❝♦♠ xy + σ(xy)(xy) xy + σ(xy)yx

  ✱ ♦✉ s❡❥❛✱ 2 2 2 αx y + ασ(xy)xyx + αxyx + ασ(xy)yx = 0, 2 y = yx ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ x ✱ ❥á q✉❡ (x, y) 6= 1✳ ❖ ♦✉tr♦ ❝❛s♦ é ❛♥á❧♦❣♦✳ x = x x −1 ∗

  ❱❛♠♦s ❛❣♦r❛ ❡st✉❞❛r ♦s ❡❧❡♠❡♥t♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞❛ ✐♥✈♦❧✉çã♦ c ✳ x = x x ∗ −1 ▲❡♠❛ ✸✳✶✷✳ ❙❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ c ∈ Z(G), ∀x ∈ G✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ x ∈ G ∗ ✱ tr✐✈✐❛❧♠❡♥t❡ t❡♠♦s ♦ r❡s✉❧t❛❞♦✳ ❙❡❥❛♠ x /∈ G ∗ ❡ y ∈ G✳ ❱❛♠♦s x , y) = 1, ♠♦str❛r q✉❡ (c ∀y ∈ G✳ −1 ∗ ∗ −1

  , y) = 1 = (x , y) = 1 x , y) = 1 ❙❡ (x, y) = 1✱ ❡♥tã♦ (x ✱ ❞♦♥❞❡ (x ✳ P♦❞❡♠♦s

  ❛❞♠✐t✐r ❡♥tã♦ q✉❡ (x, y) 6= 1✳ ∗ ∗ ∗ −1 ∗ −∗ −∗ ∗ −1 y = yx x y = x yx = yxx = yx x 2 ∗ ∗ ❙❡ xy = yx ❡ x ✱ ❡♥tã♦ x ✱ ❥á q✉❡ 2 x = (x ) ∗ ∗ ✳ ❆ss✐♠✱ s❡ y /∈ G ❡ xy = yx ♦✉ y ∈ G ✱ ♣❡❧♦ ✐t❡♠ ✭✐✐✐✮ ❞♦ ▲❡♠❛ ✸✳✻ ♦✉ ♦ ✐t❡♠ ∗ −1 x , y) = 1

  ✭✐✮ ❞♦ ▲❡♠❛ ✸✳✶✵✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❡♠♦s (x ✳ ❙✉♣♦♥❤❛ q✉❡ y /∈ G ❡ xy 6= yx ✳ P❡❧♦ ✐t❡♠ ✭✐✮ ❞♦ ▲❡♠❛ ✸✳✻✱ t❡♠♦s xy ∈

  −1 ∗ ∗ −1 ∗ ∗ ∗ ∗ ∗ ∗ −1 ∗ ∗ −∗ ✹✶ x y x = yx x y x x y = x x y = y = −∗ ∗ ∗ −1 ∗ ∗ −1 ∗ ∗ −1 ✳ ❙❡ xy = x ✱ ❡♥tã♦ yx = y ❡✱ ♣♦rt❛♥t♦✱ x yx x = y x x x , y ) = 1 x , y) = 1 ∗ −1 ✱ ♦✉ s❡❥❛✱ (x ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ (x ✳ P♦rt❛♥t♦ x x ∈ Z(G)✳

  ❖ ❧❡♠❛ ❛ s❡❣✉✐r ❡①✐❜❡✱ ❞❡ ✉♠❛ ❢♦r♠❛ ❣❡r❛❧✱ q✉❛❧ ❞❡✈❡ s❡r ❛ ❡str✉t✉r❛ ❞❛ ✐♥✈♦❧✉çã♦ ♣❛r❛ q✉❡ S s❡❥❛ ❛♥t✐❝♦♠✉t❛t✐✈♦✳

  = c x x, x ∗ ▲❡♠❛ ✸✳✶✸✳ ❙❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ x ∀x ∈ G✱ ♦♥❞❡ c ∈ G ∩ Z(G) ❡ 2 c = 1 xy = c x c y (x, y) xy x , c y , (x, y) x ✳ ❆❧é♠ ❞✐ss♦✱ c ❡ s❡ (x, y) 6= 1✱ ❡♥tã♦ c ∈ {c }✳ ,

  ❉❡♠♦♥str❛çã♦✳ Pr✐♠❡✐r♦ ✈❛♠♦s ♠♦str❛r q✉❡ (x, y) ∈ G ∀x, y ∈ G✳ ∗ ∗ ❙❡❥❛♠ x, y ∈ G t❛✐s q✉❡ (x, y) 6= 1✳ ❙❡ x /∈ G ❡ y ∈ G ✱ ❡♥tã♦✱ ♣❡❧♦ ✐t❡♠ −1 −1 ∗ y 2 x = x x ∗ ∗

  ✭✐✮ ❞♦ ▲❡♠❛ ✸✳✶✵✱ t❡♠♦s q✉❡ (x, y) = x ∈ G ✱ ❥á q✉❡ x ∈ G ❀ ♦ ❝❛s♦ x ∗ ∗ ∗ ∗ ∈ G ❡ y /∈ G é ❛♥á❧♦❣♦✳ ❙❡ x, y /∈ G ♦✉ x, y ∈ G ✱ ♣❡❧♦ ✐t❡♠ ✭✐✐✮ ❞♦ ▲❡♠❛ ✸✳✻ ♦✉ −1 −1 y /

  ❛ ❖❜s❡r✈❛çã♦ ✷✳✶✶✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❡♠♦s q✉❡ xy, x ∈ G ✱ ❛ss✐♠✱ ♣❡❧♦ ✐t❡♠ ✭✐✐✮ ❞♦ −1 −1 −1 −1 y )(xy) y ) = 1 ▲❡♠❛ ✸✳✻✱ (x, y) = (x ∈ G s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (xy, x ✱ ♣♦rt❛♥t♦✱ ❝♦♠♦ −1 −1 −1 −1 −1 −1 2 2

  (xy, x y ) = y x yxxyx y = 1 , y ✱ ❥á q✉❡ x ∈ Z(G)✱ t❡♠♦s ♦ r❡s✉❧t❛❞♦✳ 2 −1 −1 −1 −1

  = 1, y = yxy x ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ (x, y) ∀x, y ∈ G✱ ♦✉ s❡❥❛✱ xyx ✳ ❙❡❥❛♠ x, y ∈ G✳ ❙❡ (x, y) = 1✱ ♥❛❞❛ t❡♠♦s ❛ ♣r♦✈❛r✳ ❙✉♣♦♥❤❛ q✉❡ (x, y) 6= 1✳ ∗ ∗ ∗ ∗

  , y x, x y P❡❧♦s ▲❡♠❛s ✸✳✻ ❡ ✸✳✶✵✱ s❛❜❡♠♦s q✉❡ s❡ x ♦✉ y /∈ G ∗ ✱ ❡♥tã♦ xy ∈ {yx }❀ ❛❧é♠ ∗ y ∗ ∗ ❞✐ss♦✱ s❡ x, y ∈ G ✱ ❡♥tã♦ xy = x ✱ ♣♦rt❛♥t♦✱ ✈❛♠♦s ❛♥❛❧✐s❛r ❡st❛s três ♣♦ss✐❜✐❧✐❞❛❞❡s✳

  ❙❡ xy = yx ✱ ❡♥tã♦✱ ♣❡❧♦s ✐t❡♥s ✭✐✐✐✮ ❡ ✭✐✮ ❞♦s ▲❡♠❛s ✸✳✻ ❡ ✸✳✶✵✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ y t❡♠♦s yx = x ✱ ♣♦rt❛♥t♦ −1 −1 −∗ −1 xyx y = xx yy −∗

  = xx ∗ −1 ∗ 2 2 = x x (x = (x ) ) ∗ −1 −1 = x yy x −1 −1 = yxy x . x

  ❖ ❝❛s♦ xy = y é ❛♥á❧♦❣♦✳ ∗ ∗ y ❙❡ xy = x ✱ ❝♦♠♦ (x, y) ∈ G ✱ ❡♥tã♦ −1 −1 −∗ −∗ ∗ ∗ xyx y = y x y x −1 −1

  = y x yx −1 −1 −1 −1 = (yy )y (xx )x yx −2 −2 = yy xx yx −1 −1 2 2 = yxy x ,

  , y ❥á q✉❡ x ∈ Z(G)✳

  ❱❛♠♦s ❛❣♦r❛ à ♣r♦✈❛ ❞♦ ❧❡♠❛✳

  x = x x = xx = x x = c x ∗ −1 −1 −∗ ∗ −1 x ✹✷ 2 ❉❛❞♦ x ∈ G✱ t♦♠❡ c ❡ ♥♦t❡ q✉❡ c ✱ ❥á q✉❡ 2 x = 1 x = c xy xy ∗ x ∗ ∈ G ✱ ♣♦rt❛♥t♦ c ❡ c ∈ G ✳ ❚♦♠❡ y ∈ G✱ ❛ss✐♠ (xy) ❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ∗ ∗ ∗

  = y x = c y yc x x = c x c y yx xy = c x c y (y, x) = c x c y (x, y) ♣❡❧♦ ▲❡♠❛ ✸✳✶✷✱ (xy) ✱ ♦✉ s❡❥❛✱ c ✳

  ❖❜s❡r✈❡ q✉❡ ♦s ▲❡♠❛s ✸✳✻ ❡ ✸✳✶✵ ♥♦s ❛✜r♠❛♠ q✉❡ s❡ (x, y) 6= 1✱ ❡♥tã♦ ✈❛❧❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s ✐t❡♥s✿ ∗ −1 −1

  = y xy = yxy ✭❛✮ x ❀ ∗ −1 −1

  = x yx = xyx ✭❜✮ y ❀ ∗ ∗ ∗ ∗ y = x y

  ✭❝✮ xy = x ❡ xy ✳ x = x x = x y xy = (x, y) xy = (x, y)c y (x, y) = c y −1 ∗ −1 −1 ❆ss✐♠ s❡ ✭❛✮ ♦❝♦rr❡✱ ❡♥tã♦ c ❡ c ❀ y = (x, y) x = c xy ❛♥❛❧♦❣❛♠❡♥t❡✱ s❡ ✭❜✮ ♦❝♦rr❡✱ ❡♥tã♦ c ❡ c ❀ ♣♦r ✜♠✱ s❡ ✭❝✮ ♦❝♦rr❡✱ ❡♥tã♦ ∗ −1 ∗ −1 c x = x x = y y = c y xy = (x, y)

  ❡ c ✳ ▲❡♠❛ ✸✳✶✹✳ ❙✉♣♦♥❤❛ q✉❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❙❡ x, y /∈ G ✱ ❡♥tã♦✿ x = c y

  ✭✐✮ ❙❡ (x, y) = 1✱ ❡♥tã♦ c ❡ 2(1 + σ(xy)) = 0 = 2(σ(x) + σ(y))❀ ✭✐✐✮ ❙❡ (x, y) 6= 1✱ ❡♥tã♦✿ x y xy = c = c

  ✭❛✮ (x, y) = c ❡ (1 + σ(x) + σ(y) + σ(xy)) = 0✱ ♦✉❀ x y = c xy ✭❜✮ (x, y) = c 6= c ❡ σ(x) = −1✱ ♦✉❀ y x = c xy ✭❝✮ (x, y) = c ❡ σ(y) = −1✱ ♦✉❀ xy x = c y 6= c ✭❞✮ (x, y) = c 6= c ❡ σ(xy) = −1 ❡ σ(x) = −σ(y)✳

  ❉❡♠♦♥str❛çã♦✳ P❛r❛ ♠♦str❛r ✭✐✮✱ ♥♦t❡ q✉❡ s❡ (x, y) = 1 ❡ x, y /∈ G ✱ ❡♥tã♦✱ ♣❡❧♦ ▲❡♠❛ xy = c x c y x c y = 1 x = c y ✸✳✶✸✱ c ✳ P❡❧♦ ✐t❡♠ ✭✐✐✮ ❞♦ ▲❡♠❛ ✸✳✻✱ xy ∈ G ✱ ♣♦rt❛♥t♦✱ c ✱ ❧♦❣♦ c ✱ ❥á 2 2 x y = c = 1 q✉❡ c ✳ ❖ ▲❡♠❛ ✸✳✺ ❝♦♠♣❧❡t❛ ❛ ♣r♦✈❛✳

  P❛r❛ ✈❡r✐✜❝❛r ✭✐✐✮✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❛❧❣✉♥s ❝❛s♦s✳ ❙❡ x, y ∈ C✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✸✳✽✱ ❡ ❛ ❡q✉❛çã♦ ✭✸✳✶✮✱ ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛♠♦s ✭❛✮✳ ❙❡ y /∈ C✱ ❡♥tã♦ ♦ ▲❡♠❛ ✸✳✺ ❣❛r❛♥t❡ q✉❡ ✭❛✮✱ ✭❜✮ ♦✉ ✭❞✮ ♦❝♦rr❡♠✳ ❙❡ x /∈ C✱ ✉t✐❧✐③❛♥❞♦ ♦ ▲❡♠❛ ✸✳✺✱ t❡♠♦s q✉❡ ✭❛✮✱ ✭❝✮ ♦✉ ✭❞✮ ♦❝♦rr❡♠✳

  ❚❡♠♦s✱ ♥❡st❡ ♠♦♠❡♥t♦✱ ❢❡rr❛♠❡♥t❛s s✉✜❝✐❡♥t❡s ♣❛r❛ ❡st❡♥❞❡r ✉♠ ♣♦✉❝♦ ♠❛✐s ♦ ❚❡♦r❡♠❛ ✷✳✷ ❞❡ ❬●P✶✹❪✱ ♦ q✉❡ s❡rá út✐❧ ❡♠ ❜r❡✈❡✳ ◆♦ t❡♦r❡♠❛ ♦r✐❣✐♥❛❧✱ ❝❛s♦ G ❢♦ss❡ ❛❜❡❧✐❛♥♦ ❡ ∗ 6= Id✱ ♣♦✉❝♦ ♣♦❞í❛♠♦s ❝♦♥❝❧✉✐r ❛❝❡r❝❛ ❞❛ ✐♥✈♦❧✉çã♦ ❡♠ G✱ ❡①❝❡t♦ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ N = N ✳ ▼♦str❛r❡♠♦s ❛❣♦r❛ q✉❡ ❛ ✐♥✈♦❧✉çã♦ ♥♦s ❡❧❡♠❡♥t♦s ♥ã♦ s✐♠étr✐❝♦s é ❞❛❞❛ ♣r❡❝✐s❛♠❡♥t❡ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ✉♠ ❝❡rt♦ ❡❧❡♠❡♥t♦ s ∈ G ✜①♦✳ ❚❡♦r❡♠❛ ✸✳✶✺ ✭❊①t❡♥sã♦ ❚❡♦r❡♠❛ ✷✳✷✱ ❬●P✶✹❪✮✳ ❙❡❥❛ R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ t❛❧ q✉❡ charR

  6= 2 ❡ G ✉♠ ❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ∗ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ✉♠❛ ♦r✐❡♥t❛çã♦ σ : G → {±1}✳

  ✹✸ √ 2 2 R

  ✭✶✮ ⊂ R ✱ G é ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❡ ∗ = Id✳ √ 2 2

  R = R N = Id N ✭✷✮ charR = 4✱ ✱ G é ❛❜❡❧✐❛♥♦✱ ∗| ❡ ❡①✐st❡ s ∈ G t❛❧ q✉❡ x ∈ {x, sx}✳

  √ 2 2 ′ ∗ R = R =

  ) = ✭✸✮ charR = 4✱ ✱ G 2 2 = {1, s} ❡ x ∈ {x, sx} , ∀x ∈ G✳ ❆❧é♠ ❞✐ss♦✱ (G

  {1} ♦✉ R {0}✳ ∗ x = s ∗ ❉❡♠♦♥str❛çã♦✳ ❆ss✉♠✐♥❞♦ q✉❡ ∗ 6= Id✱ ✜①❡ x /∈ G ❡ ❞❡♥♦t❡ c ✳ ❉❛❞♦ y /∈ G ✱ x = c y y = s ❝♦♠♦ G é ❛❜❡❧✐❛♥♦✱ t❡♠♦s✱ ♣❡❧♦ ✐t❡♠ ✭✐✮ ❞♦ ▲❡♠❛ ✸✳✶✹✱ q✉❡ c ✱ ♣♦rt❛♥t♦ c ❡ y = ys,

  ∀y / ∈ G ✱ ❝♦♥❝❧✉✐♥❞♦ ✭✷✮✳ ❖s ❞❡♠❛✐s ❝❛s♦s ❡ ❛ r❡❝í♣r♦❝❛ s❡❣✉❡♠ ✐♥❛❧t❡r❛❞♦s✳ ❈♦♠♦ ❥á ✜③❡♠♦s ❛♥t❡r✐♦r♠❡♥t❡✱ s❛❜❡♠♦s q✉❡ ❡st❡ r❡s✉❧t❛❞♦ ❡①♣r✐♠❡ ❛ ❡str✉t✉r❛

  ❞❛ ✐♥✈♦❧✉çã♦ ❡♠ C✳ ❊st❡ t❡♦r❡♠❛ s❡rá ✉t✐❧✐③❛❞♦✱ ❡✈❡♥t✉❛❧♠❡♥t❡✱ s❡♠ q✉❛❧q✉❡r r❡❢❡rê♥❝✐❛ x = s, q✉❛♥❞♦ t♦♠❛r♠♦s c ∀x ∈ C\G ✳ ❆ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ é ❡ss❡♥❝✐❛❧♠❡♥t❡ ✉♠ r❡✜♥❛♠❡♥t♦ ❞❛ ❡str✉t✉r❛ ❞❛ ✐♥✈♦❧✉çã♦ x ,

  ❛♣r❡s❡♥t❛❞❛ ♥♦ ▲❡♠❛ ✸✳✶✸✱ ❛✜r♠❛♥❞♦ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s c ∀x ∈ G✱ ♣♦ss✉✐ ♥♦ ♠á①✐♠♦

  3 x ❡❧❡♠❡♥t♦s✳ ❆❧é♠ ❞✐ss♦✱ ♠♦str❛r❡♠♦s q✉❡ ♦s ❡❧❡♠❡♥t♦s c sã♦ ❝♦♥st❛♥t❡s ♥♦s ❝♦♥❥✉♥t♦s

  G , C ∗ ∗ ❡ G\(G ∗ \G ∩ C) ❡ q✉❡ q✉❛❧q✉❡r ❝♦♠✉t❛❞♦r ❡♠ G r❡s✉❧t❛ ❡♠ ✉♠ ❞❡st❡s ❡❧❡♠❡♥t♦s✳

  ❊st❡ r❡s✉❧t❛❞♦ ❡①✐❜✐rá ♣r❡❝✐s❛♠❡♥t❡ ❛ ❢♦r♠❛ ❞❛ ✐♥✈♦❧✉çã♦ ∗ ♣❛r❛ q✉❡ S s❡❥❛ ❛♥t✐✲ ❝♦♠✉t❛t✐✈♦✳ Pr♦♣♦s✐çã♦ ✸✳✶✻✳ ❙❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ ✈❛❧❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s✿

  ✭✐✮ G é ❛❜❡❧✐❛♥♦ ❡ ∗ = Id✳ = ✭✐✐✮ G é ❛❜❡❧✐❛♥♦ ❡ ❡①✐st❡ s ∈ G ∩ Z(G) t❛❧ q✉❡ x {x, xs} , ∀x ∈ G✳ ′ ∗ 2

  = = 1 = ✭✐✐✐✮ G {1, s} ⊂ G ∩ Z(G)✱ s ❡ x {x, xs} , ∀x ∈ G✳ C N = Id

  ✭✐✈✮ ∗| 6= Id✱ ∗| ✱ ❡①✐st❡♠ ❡❧❡♠❡♥t♦s s, t ∈ G ∩Z(G) t❛✐s q✉❡ x ∈ {x, xs} , ∀x ∈ C

  ❡ x ∈ {x, xt} , ∀x / ∈ C✳ ❆❧é♠ ❞✐ss♦✱ (x, y) ∈ {1, s, t} , ∀x, y ∈ G✱ C ⊂ {1, s}✱ (C , G) , G )) = ∗ ∗ ∗

  \G ⊂ {1, s} ❡ (C\G \(C ∪ G {s}✳ =

  ❆❧é♠ ❞✐ss♦✱ s❡ ✭✐✈✮ ♦❝♦rr❡✱ ❡♥tã♦ G {1, s}✱ C é ❛❜❡❧✐❛♥♦✱ C ⊂ Z(G)✱ (G\(C ∪ G ), G ) = ) = ∗ ∗ ∗

  {1} ❡ (G\C, C\G {s}✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ G s❡❥❛ ❛❜❡❧✐❛♥♦✳ ❙❡ ∗ = Id✱ ❡♥tã♦ ♦ ✐t❡♠ ✭✐✮ ♦❝♦rr❡✳ ❙❡ ∗ 6= Id✱ ❜❛st❛ ♣r♦❝❡❞❡r ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ à ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✸✳✶✺ ❡ ❝♦♥❝❧✉✐r ✭✐✐✮✳

  P♦❞❡♠♦s ❛❞♠✐t✐r q✉❡ G é ♥ã♦ ❛❜❡❧✐❛♥♦✳ C = Id ∗ ❙✉♣♦♥❤❛ q✉❡ ∗| ✱ ♣♦rt❛♥t♦ C ❛❜❡❧✐❛♥♦✳ ❋✐①❡ z /∈ G ✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ z / z ∗ x = s

  ∈ C✱ ❡ s❡❥❛♠ s = c ❡ x /∈ G ✳ ❆♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✸✳✶✹✱ t❡♠♦s (z, x) = 1 ❡ c ✱ x = c zx C ♦✉✱ (z, x) = s = c ✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦✱ ∗| 6= Id❀ ❛ss✐♠✱ x ∈ {x, sx} ∀x ∈ G✱ ❡

   ✹✹ (x, y) = c xy c x c y x =

  ∈ {1, s}✱ ❥á q✉❡ c ∈ {1, s} , ∀x ∈ G✱ ♦✉ s❡❥❛ G {1, s} ⊂ G ∩ Z(G)✳ ❉❡ss❛ ❢♦r♠❛✱ G ♣♦ss✉✐ ❛ ❡str✉t✉r❛ ❛♣r❡s❡♥t❛❞❛ ❡♠ ✭✐✐✐✮✳ C x

  ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ ∗| ∗ ❡ t♦♠❡ s = c ❞❛❞♦ ♣❡❧♦ ❚❡♦r❡♠❛ 6= Id✳ ❙❡❥❛ x ∈ C\G

  ✸✳✽✳ ❙❡❥❛ y ∈ G t❛❧ q✉❡ (x, y) 6= 1✳ ❙✉♣♦♥❤❛ q✉❡ y /∈ G ∗ ❀ s❡ y ∈ C✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✸✳✽✱ (x, y) = s

  ❀ s❡ y /∈ C✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✹✱ t❡♠♦s q✉❡ ✭❛✮ ♦✉ ✭❜✮ ♦❝♦rr❡♠✱ ❥á q✉❡ x ∈ C ❡ y /∈ C✱ x = s ∗ ♣♦rt❛♥t♦✱ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s (x, y) = c ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ y ∈ G ✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✵✱ ∗ xy = c x c y (x, y) x (x, y) t❡♠♦s xy ∈ G ✱ ❛ss✐♠✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✸✱ c ✱ ♦✉ s❡❥❛✱ 1 = c ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ 2

  = 1 ∗ , G) y = s ∗ ❡♠ (x, y) = s✱ ❥á q✉❡ s ✳ P♦rt❛♥t♦ (C\G ⊂ {1, s}✱ ❥á q✉❡ c ∀y ∈ C\G ✳ ∗ x = t x y = c

  ❙❡❥❛♠ x, y ∈ G\(G ∪C) ❡ c ✳ P❡❧♦ ▲❡♠❛ ✸✳✶✹✱ s❡ (x, y) = 1✱ ❡♥tã♦ t = c ❀ x = c y = (x, y) x = c y xy = (x, y) s❡ (x, y) 6= 1✱ ❡♥tã♦ t = c ✱ ♦✉✱ t = c ✱ σ(xy) = −1✱ s = c ✳ P♦rt❛♥t♦ ❛ ✐♥✈♦❧✉çã♦ ❡♠ G\C é ❞❛❞❛ ♣♦r x ∈ {x, xt = tx} , ∀x / ∈ C✳ ❚♦♠❡ ♥♦✈❛♠❡♥t❡ x ∗ ∗

  ∈ G\(G ∪ C) ❡ s❡❥❛ y ∈ G t❛❧ q✉❡ (x, y) 6= 1❀ ♥❡ss❡ ❝❛s♦✱ ♣❡❧♦s ▲❡♠❛s ✸✳✶✵ ❡ ✸✳✶✸✱ xy = c x c y (x, y) x (x, y) t❡♠♦s xy ∈ G ❡ c ✱ ♦✉ s❡❥❛✱ 1 = c ✱ ♣♦rt❛♥t♦ (x, y) = t✱ ❛ss✐♠ (G , G)

  \G ⊂ {1, s, t}✳ P♦r ✜♠✱ t♦♠❛♥❞♦ x, y ∈ G ✱ t❛✐s q✉❡ (x, y) 6= 1✱ s❛❜❡♠♦s✱ ♣❡❧❛ xy = c x c y (x, y) ❖❜s❡r✈❛çã♦ ✷✳✶✶✱ q✉❡ xy /∈ G ❡ ♥♦✈❛♠❡♥t❡ ♣❡❧♦ ▲❡♠❛ ✸✳✶✸✱ t❡♠♦s c ✱ ❧♦❣♦✱ (x, y) = c xy = s

  ♦✉ t❀ ❛ss✐♠ ❝♦♥❝❧✉í♠♦s q✉❡ G ♣♦ss✉✐ ♥♦ ♠á①✐♠♦ ❞♦✐s ❝♦♠✉t❛❞♦r❡s ♥ã♦ ∗ ∗ = = tr✐✈✐❛✐s s ❡ t❀ ❛❧é♠ ❞✐ss♦ x

  {x, xs} , ∀x ∈ C ❡ x {x, xt} ∀x / ∈ C✳ ❙❡ s = t✱ ❡♥tã♦ ❢❛❝✐❧♠❡♥t❡ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ✭✐✐✐✮ ♦❝♦rr❡✳ ❙✉♣♦♥❤❛ q✉❡ s 6= t

  ) N ❡ ♥♦t❡ q✉❡✱ ♥❡st❡ ❝❛s♦✱ G\(C ∪ G ∗ 6= ∅✳ ❙❡ ∗| 6= Id✱ ❡♥tã♦ t♦♠❛♥❞♦ y ∈ N\G ∗ ❡ x ∗ ) y = c x = t

  ∈ G\(C ∪ G ✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✹✱ t❡♠♦s q✉❡ s = c ✱ ❝♦♥tr❛❞✐çã♦✱ ❥á q✉❡ s 6= t✱ N = Id ∗ = N ∗ ∗ ) ♣♦rt❛♥t♦ ∗| ✱ ♦✉ s❡❥❛✱ N ✳ ◆♦t❡ t❛♠❜é♠ q✉❡✱ s❡ x ∈ C\G ❡ y ∈ G\(C ∪ G x = c y = t ✈❡r✐✜❝❛♠ (x, y) = 1✱ ❡♥tã♦✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✹✱ s = c ✱ ❝♦♥tr❛❞✐çã♦✱ ♣♦rt❛♥t♦ (x, y) = s , G) , G )) = ∗ ∗ ∗

  ✱ ❥á q✉❡ (C\G ⊂ {1, s}✱ ❧♦❣♦✱ (C\G \(C ∪G {s}✳ ❆ss✐♠ ❝♦♥❝❧✉í♠♦s ✭✐✈✮✳

  ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ s❡ ✭✐✈✮ é ✈❡r✐✜❝❛❞♦✱ ❡♥tã♦ G ♣♦ss✉✐ ❡①❛t❛♠❡♥t❡ ✉♠ ∗ ∗ ∗ ), G ) = ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s✱ C é ❛❜❡❧✐❛♥♦✱ C ⊂ Z(G)✱ (G\(C ∪ G {1} ❡ (G ) =

  \C, C\G {s}✳ Pr✐♠❡✐r❛♠❡♥t❡ ♠♦str❛r❡♠♦s q✉❡ (x, y) 6= t, ∀x, y ∈ G✳ ) P❛r❛ ✐ss♦✱ s✉♣♦♥❤❛ q✉❡ ❡①✐st❛♠ x, y ∈ G t❛✐s q✉❡ (x, y) = t✳ ❙❡ x ∈ G\(C ∪ G ✱ , G

  ❡♥tã♦ t♦♠❛♥❞♦ c ∈ C\G ✱ t❡♠♦s q✉❡ (x, yc) = (x, y)(x, c) = ts✱ ❥á q✉❡ (C\G \(C ∪ G )) =

  {s}✱ ❝♦♥tr❛❞✐çã♦✱ ✈✐st♦ q✉❡ G ♣♦ss✉✐ ♥♦ ♠á①✐♠♦ ❞♦✐s ❝♦♠✉t❛❞♦r❡s ♥ã♦ tr✐✈✐❛✐s s ❡ t✱ q✉❡ ♥❛t✉r❛❧♠❡♥t❡ sã♦ ❞✐st✐♥t♦s ❞❡ st❀ ❛♥❛❧♦❣❛♠❡♥t❡ ❡♥❝♦♥tr❛rí❛♠♦s ✉♠ ❛❜s✉r❞♦ ❝❛s♦ y ∗ ) ∗ ) ∗

  ∈ G\(C ∪ G ✳ P♦rt❛♥t♦✱ s❡ (x, y) = t✱ ❡♥tã♦ x, y /∈ G\(C ∪ G ✱ ♦✉ s❡❥❛✱ x, y ∈ C ∪ G ❀ ❛❧é♠ ❞✐ss♦✱ ❝♦♠♦ t /∈ C ✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ x ♦✉ y ∈ G \C✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ s✉♣♦♥❤❛✱ ∗ ∗ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ q✉❡ x ∈ G \C ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ y ∈ G ✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦ y , G) ∗ ∗

  ∈ C\G ❡✱ ❞❡ (C\G ⊂ {1, s}✱ ❝♦♥❝❧✉✐rí❛♠♦s q✉❡ (x, y) 6= t✱ ❝♦♥tr❛r✐❛♥❞♦ ❛ ❤✐♣ót❡s❡✳ ◆♦t❡ q✉❡✱ ♣❡❧❛ ❖❜s❡r✈❛çã♦ ✷✳✶✶✱ xy /∈ G ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ xy ∈ C✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦ ❡st❛rí❛♠♦s ♥❛ ♣r✐♠❡✐r❛ s✐t✉❛çã♦✱ ♦ q✉❡ é ✐♠♣♦ssí✈❡❧✳ ❆ss✐♠ xy ∈ C\G ✱ ♦ q✉❡ ❣❡r❛ ✉♠❛

  ✹✺ ❝♦♥❝❧✉í♠♦s q✉❡ G ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s✳

  ❖❜s❡r✈❡ q✉❡ ♥♦ ♣❛rá❣r❛❢♦ ❡♠ q✉❡ ❣❛r❛♥t✐♠♦s q✉❡ G ♣♦ss✉✐ ♥♦ ♠á①✐♠♦ ❞♦✐s ❝♦♠✉t❛❞♦r❡s ♥ã♦ tr✐✈✐❛✐s✱ ✐♠♣❧✐❝✐t❛♠❡♥t❡ ♠♦str❛♠♦s t❛♠❜é♠ q✉❡ ♣❛r❛ x, y ∈ G t❛✐s q✉❡ (x, y)

  6= 1✱ ✈❛❧❡♠✿ ∗ ) ∗ ✭❛✮ ❙❡ x ∈ G\(C ∪ G ❡ y ∈ G ✱ ❡♥tã♦ (x, y) = t✳ ∗ xy

  ✭❜✮ ❙❡ x, y ∈ G ✱ ❡♥tã♦ (x, y) = c ✳ P❛r❛ ✜♥❛❧✐③❛r ❛ ♣r♦✈❛✱ r❡st❛ ✈❡r✐✜❝❛r q✉❡ s❡ ✭✐✈✮ ♦❝♦rr❡✱ ❡♥tã♦ ♥❛ ✈❡r❞❛❞❡ C é ∗ ∗ ∗ ∗ ), G ) = ) =

  ❛❜❡❧✐❛♥♦✱ C ⊂ Z(G)✱ (G\(C ∪ G {1} ❡ (G\C, C\G {s}✳ ∗ ∗ ), G ) = P❡❧♦ ✐t❡♠ ✭❛✮✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ (G\(C ∪ G {1}✱ ❥á q✉❡ G ♣♦ss✉✐ ∗ ∗

  ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s ❡ ❡st❡ é ❞✐❢❡r❡♥t❡ ❞❡ t✳ ❙❡ x ∈ G \C ❡ y ∈ C ✱ ❡♥tã♦ (x, y) = 1

  ✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ ♣❡❧♦ ✐t❡♠ ✭❜✮✱ t❡rí❛♠♦s q✉❡ xy ∈ C✱ ❝♦♥tr❛❞✐çã♦✳ ❉❡ss❛ ∗ ∗ , G ❢♦r♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ (C \C) = 1✳ ◆♦t❡ q✉❡ s❡ c ∈ C ❡ x ∈ C✱ t♦♠❛♥❞♦ y /∈ C✱ t❡♠♦s q✉❡ xy /∈ C ❡✱ ♣♦rt❛♥t♦✱ (c, y) = 1 = (c, xy)✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ C ⊂ Z(G)✳ ❆❧é♠ ❞✐ss♦✱ ∗ ∗ ∗ ∗

  ❝♦♠♦ N = N ✱ s❡ x, y ∈ C\G ✱ ❡♥tã♦ x, y /∈ N ❡ ♣♦rt❛♥t♦ xy ∈ N = N ⊂ C ⊂ Z(G)✱ ❛ss✐♠ (x, y) = (x, xy) = 1✱ ❧♦❣♦ C é ❛❜❡❧✐❛♥♦✳

  P❛r❛ ✜♥❛❧✐③❛r✱ s❡❥❛♠ x ∈ G ∗ ∗ ✳ ❈❛s♦ (x, y) = 1✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✵✱ \C ❡ y ∈ C\G

  ) , G )) = t❡♠♦s q✉❡ xy ∈ G\(C ∪ G ∗ ❡ (xy, y) = 1✱ ❝♦♥tr❛❞✐çã♦✱ ❥á q✉❡ (C\G ∗ \(C ∪ G ∗ {s}✱ ∗ ∗ ) = ∗ , G

  ♣♦rt❛♥t♦ (G \C, C\G {s}✳ ❉✐ss♦ ❝♦♥❝❧✉í♠♦s t❛♠❜é♠ q✉❡ (C\G \C) = {s}✳ ❋✐♥❛❧♠❡♥t❡ ✈❛♠♦s ❛♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡ss❡ ❝❛♣ít✉❧♦✳

  ❚❡♦r❡♠❛ ✸✳✶✼✳ ❙❡❥❛ R ✉♠ ❛♥❡❧ t❛❧ q✉❡ charR 6= 2 ❡ G ✉♠ ❣r✉♣♦ ❝♦♠ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ✉♠❛ ♦r✐❡♥t❛çã♦ σ✳ S é ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✈❛❧❡♠ ♦s s❡❣✉✐♥t❡s✿

  ✭✐✮ ✭❛✮ ∗ = Id ❡ G é ❛❜❡❧✐❛♥♦❀ ♦✉ ✭❜✮ ∗ 6= Id ❡ G s❛t✐s❢❛③ ❛❧❣✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ =

  ✭✶✮ G é ❛❜❡❧✐❛♥♦ ❡ ❡①✐st❡ s ∈ G ∩ Z(G) t❛❧ q✉❡ x {x, xs} , ∀x ∈ G✳ ′ ∗ 2 = = 1 =

  ✭✷✮ G {1, s} ⊂ G ∩ Z(G)✱ s ❡ x {x, xs} , ∀x ∈ G✳ C N = Id ✭✸✮ ∗| ✱ ❡①✐st❡♠ ❡❧❡♠❡♥t♦s s, t ∈ G ∗

  6= Id✱ ∗| ∩ Z(G) t❛✐s q✉❡ x ∈ {x, xs} , ∀x ∈ C ❡ x ∈ {x, xt} , ∀x / ∈ C✳ ❆❧é♠ ❞✐ss♦✱ (x, y) ∈ {1, s, t} , ∀x, y ∈ G ∗ , G) ∗ , G ∗ )) =

  ✱ C ⊂ {1, s}✱ (C\G ⊂ {1, s} ❡ (C\G \(C ∪ G {s}✳ 2 ✭✐✐✮ ❙❡ x, y /∈ G ✱ ❡♥tã♦ xy = yx ❡ (1 + σ(xy)), (σ(x) + σ(y)) ∈ R ✱ ♦✉ xy 6= yx ❡

  (1 + σ(x) + σ(y) + σ(xy)) = 0 ✳ ∗ ∗ , y

  ✭✐✐✐✮ ❙❡ x /∈ G ∈ G ❡ α ∈ R é t❛❧ q✉❡ αy ∈ S✱ ❡♥tã♦ xy 6= yx ❡ α(1 + σ(x)) = 0✱ ♦✉ xy = yx 2 ❡ α ∈ R ✳

   ✹✻

  ✭✐✈✮ ❙❡ x, y ∈ G ❡ α, β ∈ R sã♦ t❛✐s q✉❡ αx, βy ∈ S✱ ❡♥tã♦✱ xy 6= yx ❡ αβ = 0✱ ♦✉ 2 xy = yx ❡ αβ ∈ R ✳

  ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❋❛❝✐❧♠❡♥t❡ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ Id ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ s♦♠❡♥t❡ q✉❛♥❞♦ G é ❛❜❡❧✐❛♥♦✱ ♣♦rt❛♥t♦ ✭✐✮✭❛✮ ♦❝♦rr❡✳

  ❙❡ ∗ 6= Id✱ ❛ Pr♦♣♦s✐çã♦ ✸✳✶✻ ❣❛r❛♥t❡ ✭✐✮✭❜✮✱ ♣♦rt❛♥t♦ ✈❛❧❡ ✭✐✮✳ ❖ ✐t❡♠ ✭✐✐✮ é ❣❛r❛♥t✐❞♦ ♣❡❧♦ ▲❡♠❛ ✸✳✶✸ ❡ ♦s ✐t❡♥s ✭✐✐✐✮ ❡ ✭✐✈✮ s❡❣✉❡♠ ❞♦s ▲❡♠❛s

  ✸✳✶✵ ❡ ✸✳✶✶✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ∗ −1 a = a a , a P❛r❛ ❛ r❡❝í♣r♦❝❛✱ ❞❡♥♦t❡ c ∀a ∈ G ❡ ♥♦t❡ q✉❡✱ ♣♦r ✭✐✮✱ c ∈ Z(G)✳ ❙❡❥❛♠ x, y /∈ G ❀ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ ∗ ∗ ∗ ∗

  (x + σ(x)x )(y + σ(y)y ) + (y + σ(y)y )(x + σ(x)x ) = 0, ♦✉ s❡❥❛✱ xy + yx + σ(xy)xyc x c y + σ(xy)yxc x c y + σ(y)xyc y + σ(y)yxc y + σ(x)yxc x + σ(x)xyc x ,

  ✭✸✳✷✮ s❡ ❛♥✉❧❛✳ x = c y ❙❡ (x, y) = 1✱ ❡♥tã♦✱ ♣♦r ✭✐✮✱ c ❡✱ ♣♦r ✭✐✐✮✱ 2(1+σ(xy)) = 0 = 2(σ(x)+σ(y))✱

  ♣♦rt❛♥t♦ ✭✸✳✷✮ s❡ r❡❡s❝r❡✈❡ ❝♦♠♦ xy + xy + σ(xy)xy + σ(xy)xy + σ(y)xyc x + σ(y)xyc x + σ(x)xyc x + σ(x)xyc x , ♦✉ s❡❥❛✱ 2(1 + σ(xy))xy + 2(σ(x) + σ(y))xyc x = 0. x = s = (x, y)

  ❙✉♣♦♥❤❛ q✉❡ (x, y) 6= 1✳ ❙❡ σ(x) = −1✱ ❡♥tã♦✱ ♣♦r ✭✐✮✱ c ✱ ❛ss✐♠ ✭✸✳✷✮✱ s❡ r❡❡s❝r❡✈❡ ♣♦r 2 2 xy + xys + σ(xy)xysc y + σ(xy)xys c y + σ(y)xyc y + σ(y)xysc y + σ(x)xys + σ(x)xys

  = (1 + σ(x))(xy + xys) + (σ(y) + σ(xy))(xyc y + xysc y )

  = 0; x = c y s❡ σ(y) = −1✱ ♣♦❞❡♠♦s ♣r♦✈❛r ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛❀ s❡ σ(x) 6= −1 6= σ(y)✱ ♣♦r ✭❜✮✱ c ❡ x x

  ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿ (x, y) = c ✱ ♦✉ (x, y) 6= c ✳ ❙❡ ❛ ♣r✐♠❡✐r❛ ♦❝♦rr❡✱

  ✹✼ t❡♠♦s 2 3 2 2 xy + xyc x + σ(xy)xyc + σ(xy)xyc + σ(y)xyc x + σ(y)xyc + σ(x)xyc + σ(x)xyc x x x x x =

  (1 + σ(x) + σ(y) + σ(xy))(xy + xyc x ) = x = c y 0,

  ♦♥❞❡ ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❡ ✭✐✐✮❀ s❡ c 6= (x, y)✱ ♥♦t❡ q✉❡ ❡st❛♠♦s ♥♦ ❝❛s♦ ✭✸✮ ❡ N = Id x = c y = t x = (xy) = ∗ ∗ ∗ ❛❧é♠ ❞✐ss♦✱ x, y /∈ C✱ ❥á q✉❡ ∗| ✱ ♣♦rt❛♥t♦ c ❡ yx = yttx = y xyc xy xy = (y, x) = (x, y) xy = s

  ✱ ♦✉ s❡❥❛ c 6= t✱ ❛ss✐♠ c ❡✱ ❝♦♠ ✐ss♦✱ σ(xy) = −1✱ ❥á q✉❡ N

  ⊂ G ✳ ❉❡ss❛ ❢♦r♠❛ ✸✳✷ s❡ r❡❡s❝r❡✈❡ ❝♦♠♦ xy + xys + σ(xy)xy + σ(xy)xys + σ(y)xyt + σ(y)xyst + σ(x)xyst + σ(x)xyt =

  (1 + σ(xy))(xy + xys) + (σ(y) + σ(x))(xyt + xyts) = 0,

  ❥á q✉❡ σ(xy) = −1 ❡✱ ♣♦r ✭✐✐✮✱ 1 + σ(x) + σ(y) + σ(xy) = 0✳ ∗ ∗ ) ❙❡❥❛♠ x /∈ G ❡ y ∈ G ✳ ❉❡✈❡♠♦s ♠♦str❛r q✉❡ (x + σ(x)x ❛♥t✐❝♦♠✉t❛ ❝♦♠ αy✱

  ❝♦♠ α ∈ R t❛❧ q✉❡ αy ∈ S✳ x ❙✉♣♦♥❤❛ q✉❡ (x, y) 6= 1✳ ❙❡ (x, y) 6= c ✱ ❡♥tã♦ ❡st❛♠♦s ♥♦ ❝❛s♦ ✭✸✮ ❡ ♣♦❞❡♠♦s x = t = y x = yxt ∗ ∗ ∗

  ♦❜s❡r✈❛r q✉❡ x /∈ C✱ ♦✉ s❡❥❛✱ c ✳ ❙❡ xy ∈ G ∗ ✱ ♥❡ss❡ ❝❛s♦ xy = (xy) ❡ xy xy = (xy) = y x = yxt ∗ ∗ ∗ ♣♦rt❛♥t♦ (x, y) = t✱ ❝♦♥tr❛❞✐çã♦❀ s❡ xy /∈ G ∗ ✱ ❡♥tã♦ c ✱ xy = 1, ts 2 6= 1 ❡ xyc

  ✐♠♣❧✐❝❛♥❞♦ ❡♠ (x, y) = tc ∈ {t }✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ❥á q✉❡ (x, y) 6= 1 ❡ G x

  ♣♦ss✉✐ ❛♣❡♥❛s ❞♦✐s ❝♦♠✉t❛❞♦r❡s ♥ã♦ tr✐✈✐❛✐s✳ P♦rt❛♥t♦✱ (x, y) = c ✱ ❡ ❛ss✐♠ ∗ ∗ ∗ ∗ (x + σ(x)x )αy + αy(x + σ(x)x ) = αxy + ασ(x)x y + αyx + ασ(x)yx

  = αxy + ασ(x)xyc x + αyx + ασ(x)yxc x = αxy + ασ(x)yx + αyx + ασ(x)xy = α(1 + σ(x))(xy + yx) = 0,

  ♦♥❞❡ ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❡ ✭✐✐✐✮✳ ❙❡ (x, y) = 1✱ ❡♥tã♦ xy = yx ❡ ❛♣❧✐❝❛♥❞♦ ❛ ∗ ∗ ∗ , y) = 1 )y = y(x + σ(x)x )

  ✐♥✈♦❧✉çã♦✱ t❡♠♦s q✉❡ (x ✱ ♣♦rt❛♥t♦✱ (x + σ(x)x ✱ ❧♦❣♦✱ ♣♦r ✭✐✐✐✮✱ ∗ ∗ ∗ 2 (x + σ(x)x )αy + αy(x + σ(x)x ) = 2α(x + σ(x)x )y = 0, ❥á q✉❡ α ∈ R ✳

  ❙❡❥❛♠ x, y ∈ G ❡ α, β ∈ R t❛✐s q✉❡ αx, βy ∈ S✳ P❡❧♦ ✐t❡♠ ✭✐✈✮✱ s❡❣✉❡ q✉❡ αx

  ✹✽ ❛♥t✐❝♦♠✉t❛ ❝♦♠ βy✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥❝❧✉✐♠♦s q✉❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✳

  ❱❛❧❡ r❡ss❛❧t❛r q✉❡ ♦ t❡♦r❡♠❛ ❛❝✐♠❛ ❛♣r❡s❡♥t❛ ❝♦♠♦ ❣r✉♣♦ s♦❧✉çã♦ ✉♠❛ ❝❧❛ss❡ ❞✐st✐♥t❛ ❞❛s ❥á ❡♥❝♦♥tr❛❞❛s ♣❛r❛ ✐❞❡♥t✐❞❛❞❡s tr❛t❛❞❛s ❛té ♦ ♠♦♠❡♥t♦✱ ♣❡r♠✐t✐♥❞♦ q✉❡ ♦ ❣r✉♣♦ ♣♦ss✉❛ ❛té ❞♦✐s ❝♦♠✉t❛❞♦r❡s ♥ã♦ tr✐✈✐❛✐s ❡ q✉❡ ❛ ✐♥✈♦❧✉çã♦ ♥♦s ❡❧❡♠❡♥t♦s ♥ã♦ s✐♠étr✐❝♦s ♣♦ss❛ s❡r ❞❛❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❞♦✐s ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s s ❡ t✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ s❡❣✉✐♥t❡ ❝♦r♦❧ár✐♦ ✐rá ♠♦str❛r q✉❡ ♥❛ ❝❧❛ss❡ ❞❡s❝r✐t❛ ❛❝✐♠❛✱ ♦ ❣r✉♣♦ ♣♦ss✉✐ ♥❛ ✈❡r❞❛❞❡ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧✱ s❡♥❞♦ ❛ss✐♠✱ ♦ ❢❛t♦ q✉❡ ♦ ❞✐st✐♥❣✉❡ ❞♦s ❞❡♠❛✐s é ❛ ✐♥✈♦❧✉çã♦ s❡r ❞❛❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛✐s ❞❡ ✉♠ ❡❧❡♠❡♥t♦ ♥♦ ❝❛s♦ ❞♦s ♥ã♦ s✐♠étr✐❝♦s✳

  ❆❧é♠ ❞❡ ❞❛r ♠❛✐s ♣r❡❝✐sã♦ ❛ ❡str✉t✉r❛ ❞❡ G✱ ♦ r❡s✉❧t❛❞♦ ❛ s❡❣✉✐r ♥♦s ❢♦r♥❡❝❡ ♠❛✐s ✐♥❢♦r♠❛çõ❡s ❛❝❡r❝❛ ❞❡ R✱ ❞❡ G/N✱ ♠♦str❛ ❛❧❣✉♠❛s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ s ∈ N✱ ❛❧é♠ ♠♦str❛r ❝♦♠ ♣r❡❝✐sã♦ ❛ tá❜✉❛ ❞❡ ❝♦♠✉t❛❞♦r❡s ❞❡ G✳ ❈♦r♦❧ár✐♦ ✸✳✶✽✳ ◆❛s ❝♦♥❞✐çõ❡s ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ t❡♠♦s✿

  √ 2 2 2 R ) ✭■✮ ❙❡ ∗ = Id✱ ❡♥tã♦ ( ⊂ R ❡ exp(G/N) = 2❀

  √ 2 2 R = R ✭■■✮ ❙❡ ∗ 6= Id✱ ❡♥tã♦ charR = 4✱ ❡ exp(G/N) = 2 ♦✉ 4✳

  ✭■■■✮ ❙❡ ∗ 6= Id ❡✱ ❛❞✐❝✐♦♥❛❧♠❡♥t❡✱ C\G ∗ 6= ∅✱ exp(G/N) = 2 ♦✉ G é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ s

  ∈ N✳ ✭■❱✮ ❙❡ ✭✸✮ ♦❝♦rr❡✱ ♦s ❝♦♠✉t❛❞♦r❡s ❞❡ G sã♦ ❞❡s❝r✐t♦s ♣❡❧❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✱ ❝✉❥❛s ❡♥tr❛❞❛s sã♦ ❞❛❞❛s ♣❡❧♦s ♣♦ssí✈❡✐s ❝♦♠✉t❛❞♦r❡s ❞❡ ❡❧❡♠❡♥t♦s ❞❛s r❡s♣❡❝t✐✈❛s ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s✿

  C C G ) ∗ ∗ ∗ ∗ \G \C G\(C ∪ G

  C ∗ ✶ ✶ ✶ ✶ C ∗

  \G ✶ ✶ s s G

  \C ✶ s ✶✱s ✶ G )

  \(C ∪ G ✶ s ✶ ✶✱s ∗ ∗ ❆❧é♠ ❞✐ss♦✱ ♥ã♦ é ♣♦ssí✈❡❧ ♠❡❧❤♦r❛r ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❡♠ (G \C, G \C) ❡ ❡♠ (G\(C∪ G ), G )) ∗ ∗

  \(C ∪ G ✳ √ 2 R 2 )

  ❉❡♠♦♥str❛çã♦✳ ❖❜s❡r✈❡ q✉❡ ♦ ▲❡♠❛ ✸✳✶ ❣❛r❛♥t❡ q✉❡ s❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ ( ⊂ √

  R 2 R 2 = R 2 ✱ ❝❛s♦ ∗ = Id❀ charR = 4 ❡ ✱ ❝❛s♦ ∗ 6= Id✳

  ❖s r❡s✉❧t❛❞♦s ❛❝❡r❝❛ ❞❡ exp(G/N) s❡❣✉❡♠ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ ✐t❡♠ ✭■■✮ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✹✳ ∗ ∗ −1 ∗ −1 ∗ ∗ P❛r❛ ✈❡r✐✜❝❛r ♦ ✐t❡♠ ✭■■■✮✱ ♥♦t❡ q✉❡✱ s❡ C\G 6= ∅✱ t♦♠❛♥❞♦ x ∈ C\G ✱ ❡♥tã♦ s = x x x ) = σ(xx ) = 1

  ❡✱ ♣♦rt❛♥t♦ σ(s) = σ(x ✱ ♦♥❞❡ ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ 2 = 1

  ❞❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❞❡ σ ❝♦♠ ∗ ❡ ❛ ♣❡♥ú❧t✐♠❛ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ σ(x) ✱ ❥á q✉❡ x ∈ C❀

  ✹✾ ❛♦ ❝❛s♦ ❛♥t❡r✐♦r❀ s❡ G é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ ❡①✐st❡♠ x, y ∈ G t❛✐s q✉❡ s = (x, y)✱ ❛ss✐♠ −1 −1 σ(s) = σ((x, y)) = σ(x )σ(y )σ(x)σ(y) = 1

  ❞❡✈✐❞♦ à ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ R✳ ij ❱❛♠♦s ♠♦str❛r ✭■❱✮✳ P❛r❛ ✐ss♦✱ ❞❡♥♦t❛r❡♠♦s ♣♦r a ❛s ❡♥tr❛❞❛s ❞❛ i✲és✐♠❛ ❧✐♥❤❛

  ❞❛ j✲és✐♠❛ ❝♦❧✉♥❛ ❞❛ t❛❜❡❧❛✱ ♦❜✈✐❛♠❡♥t❡ ❞❡s❝♦♥s✐❞❡r❛♥❞♦ ❛s ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s q✉❡ ❞❡t❡r♠✐✲ ♥❛♠ ♦s ❝♦♥❥✉♥t♦s ❛ s❡ ❝♦❧♦❝❛r ♥♦s ❝♦♠✉t❛❞♦r❡s✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ ❡ss❛ t❛❜❡❧❛ é s✐♠étr✐❝❛ ij ❡♠ r❡❧❛çã♦ à ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧✱ é s✉✜❝✐❡♥t❡ ❞❡t❡r♠✐♥❛r ♦s a ❝♦♠ i ≤ j✳

  ◆♦t❡ q✉❡✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✸✳✶✻✱ ❡st❛♠♦s ♥❛ ❝♦♥❞✐çã♦ ✭✐✈✮ ❡✱ ♣♦rt❛♥t♦✱ ❞❡ C ⊂ 1 j ∗ ) = 22 Z(G) ❡ C ❛❜❡❧✐❛♥♦✱ ❥á t❡♠♦s ❞❡s❝r✐t♦ ♦s ❡❧❡♠❡♥t♦s a ❡ a ❀ ❞❡ (G\C, C\G {s}✱ t❡♠♦s a 23 24 ), G ) = ∗ ∗ 34 33 44 ❡ a ❀ ❞❡ (G\(C ∪ G {1} t❡♠♦s a ✳ ❘❡st❛ ❡♥tã♦ ✈❡r✐✜❝❛r q✉❡ a ❡ a ♥ã♦

  ♣♦❞❡♠ s❡r ♠❡❧❤♦r❛❞❛s ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ t❛♥t♦ 1 q✉❛♥t♦ s sã♦ r❡s✉❧t❛❞♦s ❞❡ ❝♦♠✉t❛❞♦r❡s ♥❡st❡s ❝♦♥❥✉♥t♦s✳

  ❉❡ ❢❛t♦✱ ❡♠ ❛♠❜❛s é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠ ❝♦♠✉t❛❞♦r r❡s✉❧t❛♥❞♦ ❡♠ 1 ❥á q✉❡ ♥❡ss❡s ❝❛s♦s ❡st❛♠♦s tr❛t❛♥❞♦ ❞♦ ❝♦♠✉t❛❞♦r ❞❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❝♦♥s✐❣♦ ♣ró♣r✐♦❀ ♣♦r ∗ ∗ ♦✉tr♦ ❧❛❞♦✱ ♥♦t❡ q✉❡ t♦♠❛♥❞♦ x ∈ G \C ❡ y ∈ C\G ✱ t❡♠♦s q✉❡ (x, y) = s✱ ❧♦❣♦✱ ♣❡❧♦ ▲❡♠❛ ✸✳✶✵✱ xy ∈ G \C ❡ ❛❧é♠ ❞✐ss♦ (x, xy) = (x, y) = s✱ ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ 33

  (G ∗ ∗ \C, G \C)∩{s} 6= ∅✱ ♦✉ s❡❥❛✱ ❡①✐st❡♠ ❛s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❡♠ a ❀ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ 44 t♦♠❛♥❞♦ ❛❣♦r❛ x ∈ G\(G ∗

  ✳ ∪C) ❡ ✉t✐❧✐③❛♥❞♦ ♦ ▲❡♠❛ ✸✳✻✱ ❝♦♥❝❧✉í♠♦s ♦ ♠❡s♠♦ ♣❛r❛ a

  P♦r ✜♠✱ ❛ s❡❣✉✐r t❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❣r✉♣♦ q✉❡ ✈❡r✐✜❝❛ ❛ ♥♦✈❛ ❝❧❛ss❡ ❝✐t❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡ ❡ q✉❡ s❡ ❡♥❝❛✐①❛ ♥❛s ❝♦♥❞✐çõ❡s ❞♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❞❡st❡ ❝❛♣ít✉❧♦✳ ❊①❡♠♣❧♦ ✸✳✶✾✳ ❙❡❥❛♠ 4 4 2 2 G = x, y, g : x = y = g = (x, y) = (g, y) = 1, (x, g) = x ,

  ♦✉ s❡❥❛ 4 4 2 G = ( ) ⋊ C 4 4 hxi × hyi) ⋊ hgi ≃ (C × C ❡ R = Z × Z ✳ ❉❡✜♥❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❡♠ G t♦♠❛♥❞♦ ❛ ❡①t❡♥sã♦ ❛♦ ♣r♦❞✉t♦ ❛ ♣❛rt✐r ∗ ∗ ∗ 2 2

  = xx = xs = y = gy = gt ❞❡ x ✱ y ❡ g ✱ ❡ ✉♠❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛ σ ❛ ♣❛rt✐r ❞❡ σ(x) = (3, 3)✱ σ(y) = (1, 1) ❡ σ(g) = (1, 3)✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ ❡ss❛ ♦r✐❡♥t❛çã♦ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦ ∗✱ ♣♦rt❛♥t♦ σ∗ ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛ 4 4

  )G ❡♠ (Z ✳

  × Z ❋❛❝✐❧♠❡♥t❡ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ C é ❛❜❡❧✐❛♥♦✱ G ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r 2

  , y ∗ = C ∗ = G ∗ = ♥ã♦ tr✐✈✐❛❧ s✱ ❡ x ∈ {x, sx} , ∀x ∈ C✳ ❆❧é♠ ❞✐ss♦✱ N = hx i = N Z(G)✱ C = x ∗ = , x y , xy s 3 i 3 i i i ❡ ∗| 6= Id✱ ❥á q✉❡ x ✳ ◆♦t❡ q✉❡ C\G {xy } = {xy } ❡ G\C = gC✱ ♣♦rt❛♥t♦ s = (x, g) = (x, gc) = (xs, gc) = (xs, g), ∀c ∈ C✱ ❥á q✉❡ (c, x) = 1 ❡ s ∈ Z(G)✱ ❛❧é♠ ❞✐ss♦ ♦s ❝♦♠✉t❛❞♦r❡s ❛♥t❡r✐♦r❡s ♥ã♦ s♦❢r❡♠ i , G ❛❧t❡r❛çã♦ q✉❛♥❞♦ s✉❛s ❡♥tr❛❞❛s sã♦ ♠✉❧t✐♣❧✐❝❛❞❛s ♣♦r y ∈ Z(G)✱ ♣♦rt❛♥t♦ (C\G \C) =

  2 ✺✵ ∗ ∗ ∗ ∗ = , y = xC

  ❈♦♠♦ Z(G) = C hx i ❡ C\G ✱ s❡ c ∈ C ✱ ❡♥tã♦ ∗ ∗ ∗ (cg) = g c = gtc = cgt, ∗ ∗ ∗

  (xcg) = (cg) x = cgtxs = gxcts = xgscts = xcgt, ♦✉ s❡❥❛✱ ❛ ✐♥✈♦❧✉çã♦ ❡♠ G\C é ❞❛❞❛ ♣♦r a ∈ {a, at} , ∀a ∈ G\C✳ ❉❡ss❛ ❢♦r♠❛ ❝♦♥❝❧✉í♠♦s q✉❡ G ✈❡r✐✜❝❛ ❛s ❝♦♥❞✐çõ❡s ❞♦ ✐t❡♠ ✭✐✮✭❜✮✭✸✮ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳

  P❛r❛ ✈❡r✐✜❝❛r ✭✐✐✮✱ ♥♦t❡ q✉❡ ♦s ♣♦❧✐♥ô♠✐♦s✱ ♥❛s ✈❛r✐á✈❡✐s u ❡ v✱ 2(1+uv), 2(u+v) 4 4 ) ❡ (1+u+v +uv) sã♦ ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦s ❡♠ U(Z ×Z ✱ ❛ss✐♠✱ ♦ ✐t❡♠ ✭✐✐✮ é tr✐✈✐❛❧♠❡♥t❡ ✈❡r✐✜❝❛❞♦✳ = 2 ❈♦♠♦ G Z(G)✱ ♣❛r❛ ✈❡r✐✜❝❛r ♦s ✐t❡♥s ✭✐✐✐✮ ❡ ✭✐✈✮ é s✉✜❝✐❡♥t❡ ✈❡r✐✜❝❛r q✉❡

  α , ∈ R ∀α ∈ R t❛❧ q✉❡ αa ∈ S ❝♦♠ a ∈ G ✳ ◆♦t❡ q✉❡ ❡ss❛ ❝♦♥❞✐çã♦ t❛♠❜é♠ é

  √ 2 2 ∗ ∗ ∗ = N R = R

  ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛❞❛✱ ✈✐st♦ q✉❡ G ❡ s❡ αa ∈ S ❝♦♠ a ∈ N ✱ ❡♥tã♦ α ∈ ✱ 1 ♣♦✐s ❡ss❛ é ❡①❛t❛♠❡♥t❡ ❛ ❝♦♥❞✐çã♦ ♣❛r❛ q✉❡ αa ∈ S ✳

  ❉❡ss❛ ❢♦r♠❛✱ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✸✳✶✼✱ ❣❛r❛♥t✐♠♦s q✉❡ S é ❛♥t✐❝♦♠✉t❛t✐✈♦✳

  ❈❛♣ít✉❧♦ ✹ ❈♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❆♥t✐ss✐♠étr✐❝♦s

  ❈♦♠♦ ❥á ❡①✐❜✐♠♦s ♥♦ ❈❛♣ít✉❧♦ ✶✱ ♦ ❡st✉❞♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✈✐✲ + ❞❛❞❡ ❞❡ RG ❡ RG ❡stá✱ ♥♦ s❡♥t✐❞♦ ❛❜♦r❞❛❞♦ ♥❡st❛ t❡s❡✱ ❝♦♥❝❧✉í❞♦ ♣❛r❛ ❛s ✐♥✈♦❧✉çõ❡s ♥ã♦ ♦r✐❡♥t❛❞❛s ❡ q✉❛s❡ ❝♦♠♣❧❡t♦ ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ✉s✉❛❧✱ r❡st❛♥❞♦ ❛♣❡✲ ♥❛s ❝❛r❛❝t❡r✐③❛r ♦ q✉❡ ♦❝♦rr❡ q✉❛♥❞♦ RG é ❝♦♠✉t❛t✐✈♦ ❡ ∗ é ✉♠❛ ✐♥✈♦❧✉çã♦ q✉❛❧q✉❡r ❡♠ G

  ✳ ❉❡♥tr❡ ❛s ✐❞❡♥t✐❞❛❞❡s ❛ss✉♠✐❞❛s ♣❛r❛ ♦s ❝♦♥❥✉♥t♦s ❞♦s s✐♠étr✐❝♦s✱ ♣♦❞❡♠♦s ♣❡r✲

  ❝❡❜❡r q✉❡ ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❛♣r❡s❡♥t❛ ❝♦♠♦ s♦❧✉çõ❡s ❣r✉♣♦s ❝♦♠ ✐♥✈♦❧✉çõ❡s ❞❡ ❡s✲ tr✉t✉r❛s ♠✉✐t♦ s✐♠✐❧❛r❡s✱ ❛♣❛r❡❝❡♥❞♦ s❡♠♣r❡ ❝♦♠♦ s♦❧✉çõ❡s ❣r✉♣♦s ❝♦♠ ♥♦ ♠á①✐♠♦ ✉♠ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ ❡ ✐♥✈♦❧✉çõ❡s ❞❛❞❛s✱ ♥♦s ❡❧❡♠❡♥t♦s ♥ã♦ s✐♠étr✐❝♦s✱ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛✲ çã♦ ♣♦r ✉♠ ❡❧❡♠❡♥t♦ ✜①♦✱ ❡①❝❡t♦ ❡♠ ✉♠ ❞♦s ❝❛s♦s ❞♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r q✉❡ ❛♣r❡s❡♥t❛♠ ❞♦✐s ❡❧❡♠❡♥t♦s✱ s❡♥❞♦ q✉❡ ❡st❡s ❡❧❡♠❡♥t♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ♣♦ss✉❡♠ ✉♠❛ ❢♦rt❡ r❡❧❛çã♦ ❝♦♠ + ♦ s✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦✱ ❝❛s♦ ❡st❡ s❡❥❛ ♥ã♦ tr✐✈✐❛❧✳ ❏á ♥♦ ❡st✉❞♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡✱ ♣❛r❛ RG

  ✱ ❡♠❜♦r❛ ♦ ❝❛s♦ ♥ã♦ ♦r✐❡♥t❛❞♦ s❡❥❛ ♠✉✐t♦ ❜❡♠ ❝♦♠♣♦rt❛❞♦✱ ❡①✐❜✐♥❞♦ ❝♦♠♦ s♦❧✉çõ❡s ❣r✉♣♦s ❙▲❈s✱ ❛♦ ✐♥tr♦❞✉③✐r♠♦s ✉♠❛ ♦r✐❡♥t❛çã♦ ✭✉s✉❛❧✮✱ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ✈ã♦ ❛❧é♠ ❞❡ss❡s ❝❛s♦s❀ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ tr❛t❛r ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ RG s❡ ❛♣r❡s❡♥t❛ ❞❡s❞❡ ♦ ❡st✉❞♦ ❞❡ ✐♥✈♦❧✉çõ❡s ♥ã♦ ♦r✐❡♥t❛❞❛s✱ ✈✐st♦ q✉❡ ❢♦✐ ♥❡❝❡ssár✐♦ ✉♠ ❡s❢♦rç♦ ♠❛✐♦r ♣❛r❛ ❝♦♥❝❧✉✐r ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ss❡ ❝❛s♦✱ ✈✐❞❡ ❬❏❘✵✺✱ ❇❏P❘✵✾❪✱ ❡ ❛❧é♠ ❞✐ss♦✱ ♦s ❣r✉♣♦s s♦❧✉çõ❡s ♥ã♦ sã♦ tã♦ s✐♠♣❧❡s ❞❡ s❡ tr❛t❛r ❝♦♠♦ ♥❛s s✐t✉❛çõ❡s ❛♥t❡r✐♦r❡s✳

  ❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ❝❛r❛❝t❡r✐③❛ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ RG ❡ t♦r♥❛ ❡✈✐❞❡♥t❡ ❛ ❞✐st✐♥çã♦ ❞❡st❡ ❝❛s♦ ♣❛r❛ ♦s ❞❡♠❛✐s✳ ❚❡♦r❡♠❛ ✹✳✶ ✭❚❡♦r❡♠❛ ✷✳✺✱ ❬❇❏P❘✵✾❪✮✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ t❛❧ q✉❡ charR 6= 2 ❡ ∗ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ G✳ ❖ ❝♦♥❥✉♥t♦ RG é ❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ♦❝♦rr❡✿

  ✭✶✮ G é ❛❜❡❧✐❛♥♦✳ ✭✷✮ K = hx ∈ G : x /∈ G ∗ ∗ ❡ ∗ −1 i é ❛❜❡❧✐❛♥♦ ✭❝♦♥s❡q✉❡♥t❡♠❡♥t❡ G = K ∪ Ky✱ ♦♥❞❡ y ∈ G 2

  ′ ∗ 2 ✺✷

  = = 1 = ✭✸✮ charR = 4✱ G {1, s} ⊂ Z(G)✱ s ❡ x {x, xs} , ∀x ∈ G✳ ❆❧é♠ ❞✐ss♦✱ G é 2 2

  ❝♦♠✉t❛t✐✈♦ ❝❛s♦ R 6= {0}✳ 2 =

  ✭✹✮ R ∗ ✳ {0} ❡ G ❝♦♥té♠ ✉♠ s✉❜❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❞❡ í♥❞✐❝❡ 2 q✉❡ ❡stá ❝♦♥t✐❞♦ ❡♠ G ′ ′ ′ 3

  = = (G/G ) ✭✺✮ charR = 3✱ G {1, s, t}✱ G/G ∗ ❡ x ∈ G ∗ ∀x ∈ G✳

  ❆ss✐♠ ❝♦♠♦ ♥♦s ❝❛s♦s ❛♥t❡r✐♦r❡s✱ ❡ss❡ t❡♦r❡♠❛ ♥♦s ❛ss❡❣✉r❛ ❛ ❡str✉t✉r❛ ❞♦ s✉❜✲ ❣r✉♣♦ N ≤ G✳ ❱✐st♦ ❛ ❣r❛♥❞❡ ✈❛r✐❡❞❛❞❡ ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ N ❡ ❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ tr❛t❛r − −

  = RG ❛❧❣✉♥s ❞❡❧❡s✱ ❡①✐❜✐r❡♠♦s ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ ♣❛r❝✐❛❧ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ RG σ ∗ ✱ s❡♥❞♦ σ : G → {±1}✳

  ❉✉r❛♥t❡ t♦❞♦ ♦ ❝❛♣ít✉❧♦✱ tr❛t❛r❡♠♦s ❞❡ ✉♠❛ ♦r✐❡♥t❛çã♦ ✉s✉❛❧ ♥ã♦ tr✐✈✐❛❧✱ ♦✉ s❡❥❛✱ σ : G

  → {±1}✳

  ✹✳✶ R

  2

  6= {0} ♦✉ (G ∩ Z(G))\N 6= ∅

  ∗ ❚r❛t❛r❡♠♦s ❛ ♣❛rt✐r ❞❡ ❛❣♦r❛ ♦ q✉❡ ♦❝♦rr❡ ❝❛s♦ ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ❤✐♣ót❡s❡s 2

  ♦❝♦rr❛♠✱ R 6= {0} ♦✉ (G ∩ Z(G))\N 6= ∅✳ P❡❧❛ ❞❡s❝r✐çã♦ ❞♦s ❣❡r❛❞♦r❡s ❞❡ RG ❛♣r❡s❡♥t❛❞❛ ♥♦ ❈❛♣ít✉❧♦ ✶✱ t❡♠♦s q✉❡ ❡st❡

  é ❣❡r❛❞♦ ♣♦r 2 : x / {x − σ(x)x ∈ G ∗ } ∪ {rx : x ∈ N ∗ ❡ r ∈ R } ∪ G ∗ \N.

  ❆ ✈❛♥t❛❣❡♠ ❡♠ s❡ t♦♠❛r t❛✐s r❡str✐çõ❡s✱ s❡ ❡♥❝♦♥tr❛ ♥♦ ❢❛t♦ ❞❡ q✉❡ ❞❡ ✉♠❛ ❢♦r♠❛ 2 ❞✐r❡t❛✱ ♥♦ ❝❛s♦ ❞❡ R 6= {0}✱ ♦✉ ✐♥❞✐r❡t❛✱ (G ∩ Z(G))\N 6= ∅✱ ❝♦♥s❡❣✉✐♠♦s ❝♦❧♦❝❛r ❡❧❡♠❡♥t♦s ❞❡ G ❡♠ ❝♦♠✉t❛❞♦r❡s✳ ❖ ❧❡♠❛ ❛ s❡❣✉✐r ❡✈✐❞❡♥❝✐❛rá ❝♦♠♦ ♣♦❞❡♠♦s ❢❛③❡r ✐ss♦ ♥♦ ❝❛s♦ (G ∩ Z(G))\N 6= ∅✳ ∗ ∗ ) , RG ] = 0

  ▲❡♠❛ ✹✳✷✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❝♦♠✉t❛t✐✈♦✳ ❙❡ (Z(G)∩G \N 6= ∅✱ ❡♥tã♦ [G ✳ − − ] = 0

  ❉❡♠♦♥str❛çã♦✳ ❙❡ x ∈ G \N✱ ❡♥tã♦ x ∈ RG ✱ ♣♦rt❛♥t♦ ♣♦r ❤✐♣ót❡s❡ [x, RG ✳ ∗ ∗ ∗ ❙❡ x ∈ N ✱ s❡❥❛ z ∈ (G ∩ Z(G))\N ❡ ♥♦t❡ q✉❡ zx ∈ G \N✱ ♣♦rt❛♥t♦ zx ∈ RG ✱ ❛ss✐♠ ❞❛❞♦ m ∈ RG ✱ t❡♠♦s z[x, m] = [zx, m] = 0, ♦✉ s❡❥❛ [x, m] = 0. 2

  ▲❡♠❛ ✹✳✸✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❝♦♠✉t❛t✐✈♦ ❡ σ : G → {±1} é ♥ã♦ tr✐✈✐❛❧✳ ❙❡ R ∗ ∗ 6= {0} )

  = x x, ♦✉ (Z(G) ∩ G ∗ \N 6= ∅✱ ❡♥tã♦ G ∗ ⊂ Z(G)✳ ❊♠ ♣❛rt✐❝✉❧❛r xx ∀x ∈ G✳ 2

  ❉❡♠♦♥str❛çã♦✳ ❈❛s♦ ✶✿ R 6= {0}✳ − − ❙❡❥❛♠ x ∈ G \N✱ ❧♦❣♦ x ∈ RG ✱ ❡ y ∈ G✳ ❙❡ y ∈ G ✱ ry ∈ RG ✱ ❝♦♠ r 2

  ∈ R \ {0}✱ ❡ ♣♦rt❛♥t♦ r[x, y] = [x, ry] = 0✱ ✐♠♣❧✐❝❛♥❞♦ ❡♠ (x, y) = 1✳ ❙❡ y / ∈ G ✱ ❡♥tã♦

  ✺✸ ♦✉ s❡❥❛✱ ∗ ∗ xy + σ(y)y x = yx + σ(y)xy , x

  ♣♦rt❛♥t♦✱ ❝♦♠♦ y /∈ G ❡ charR 6= 2✱ xy = yx ♦✉ xy = y ❡ y /∈ N✱ ❛ss✐♠✱ s❡ ❛ s❡❣✉♥❞❛ ∗ ∗ x = (xy) ♦♣çã♦ ♦❝♦rr❡r✱ xy = y ✱ ♣♦rt❛♥t♦ xy ∈ G ∗ ✱ ❞❡ss❛ ❢♦r♠❛✱ ♣❡❧♦ ❝❛s♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ s❛❜❡♠♦s q✉❡ (x, y) = (x, xy) = 1✳ ❆ss✐♠✱ G ∗ ∗ , G ∗ \N ⊂ Z(G)✳

  ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ (N \N) = 1✳ P❛r❛ ✐ss♦✱ s❡❥❛ n ∈ N ❡ s✉♣♦♥❤❛ q✉❡ ❡①✐st❛ x /∈ N t❛❧ q✉❡ (n, x) 6= 1✳ P❡❧♦ ♣r♦✈❛❞♦ ❛❝✐♠❛✱ s❛❜❡♠♦s q✉❡ x /∈ G ✱ ♣♦rt❛♥t♦✱ 2 t♦♠❛♥❞♦ r ∈ R \ {0}✱ s❛❜❡♠♦s q✉❡

  [rn, x + x ] = 0, ♦✉ s❡❥❛✱ ∗ ∗ r(nx + nx n) = 0,

  − xn − x n ♣♦rt❛♥t♦✱ ❝♦♠♦ x /∈ G ✱ s❛❜❡♠♦s q✉❡ nx = xn ♦✉ nx = x ✱ ❛❧é♠ ❞✐ss♦ nx /∈ N✳ ▲♦❣♦ s❡ ♦ s❡❣✉♥❞♦ ❝❛s♦ ♦❝♦rr❡ nx ∈ G \N ⊂ Z(G)✱ ❡ ♣♦rt❛♥t♦ (n, x) = (n, nx) = 1✱ ❝♦♥tr❛❞✐çã♦✱

  , G ❧♦❣♦✱ (N ∗

  \N) = 1✳ , N ) = 1

  ❘❡st❛ ✈❡r✐✜❝❛r q✉❡ (N ∗ ✳ ❙❡ n ∈ N ∗ ❡ x ∈ N✱ t♦♠❛♥❞♦ y /∈ N✱ s❛❜❡♠♦s q✉❡ xy /∈ N✱ ❛ss✐♠✱ ♣❡❧♦ ❝❛s♦ ❛♥t❡r✐♦r✱ (n, y) = 1 = (n, xy)✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ (n, x) = 1✳ ∗ ) ❈❛s♦ ✷✿ (Z(G) ∩ G \N 6= ∅✳ ∗ ∗ ) ❙❡❥❛♠ x ∈ G ✱ y ∈ G ❡ z ∈ (Z(G) ∩ G \N✳ ❙✉♣♦♥❤❛ q✉❡ y ∈ G ✱ ❡ ♥♦t❡ q✉❡ y ♦✉ zy /∈ N✱ ♣♦rt❛♥t♦✱ ♣❡❧♦ ▲❡♠❛ ✹✳✷✱ t❡♠♦s q✉❡ [x, y] = 0 ♦✉ z[x, y] = [x, zy] = 0✱ ❛ss✐♠ (x, y) = 1✳

  ] = 0 ❙❡ y /∈ G ✱ ♥♦✈❛♠❡♥t❡ ♣❡❧♦ ▲❡♠❛ ✹✳✷✱ [x, y − σ(y)y ✱ ♦✉ s❡❥❛ ∗ ∗ xy + σ(y)y x = yx + σ(y)xy , ∗ ∗ ∗ x x = (xy)

  ♣♦rt❛♥t♦ ❝♦♠♦ y /∈ G ∗ ✱ xy = yx ♦✉ xy = y ✳ ❙❡ xy = y ✱ ❡♥tã♦ xy ∈ G ∗ ❡✱ ♣❡❧♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✱ (x, y) = (x, xy) = 1✱ ❝♦♥❝❧✉✐♥❞♦ ♦ ❈❛s♦ ✷✳ ∗ ∗ ∗

  ) = 1 ∗ ) = P❛r❛ ♠♦str❛r q✉❡ (x, x é s✉✜❝✐❡♥t❡ ✈❡r✐✜❝❛r q✉❡ xx ∈ G ✱ ♣♦✐s (x, x (x, xx ) = 1

  ✳ 2 ❆ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ ❛s ❤✐♣ót❡s❡s R 6= {0} ♦✉ (G ∩ Z(G))\N 6= ∅ s❡rã♦ ❛ss✉✲

  ♠✐❞❛s ❛té ♦ ✜♥❛❧ ❞♦ ❝❛♣ít✉❧♦ ❡✱ ♣♦rt❛♥t♦✱ ❛s ♠❡s♠❛s s❡rã♦ ♦♠✐t✐❞❛s ❞♦s ❡♥✉♥❝✐❛❞♦s ❞♦s r❡s✉❧t❛❞♦s✳ ▲❡♠❛ ✹✳✹✳ ❙✉♣♦♥❤❛ q✉❡ RG s❡❥❛ ❝♦♠✉t❛t✐✈♦✳ ❉❛❞♦s x, y ∈ G✱ ❡♥tã♦ xy = yx s❡✱ ❡ ∗ ∗ y = yx s♦♠❡♥t❡ s❡✱ x ✳ ❉❡♠♦♥str❛çã♦✳ ❆ ♣r♦✈❛ é ❛♥á❧♦❣❛ ❛♦ ▲❡♠❛ ✷✳✹✱ ✈✐st♦ q✉❡ ♣❛r❛ ❛ ♣r♦✈❛ ❞❡ss❛ ú❧t✐♠❛ é ∗ ∗

  = x x,

  ✺✹ ❆ss✐♠ ❝♦♠♦ ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✱ ✉t✐❧✐③❛r❡♠♦s ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ s❡♠ ❢❛③❡r r❡❢❡rê♥❝✐❛ ❛♦ ❧❡♠❛✳

  ▲❡♠❛ ✹✳✺✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❝♦♠✉t❛t✐✈♦ ❡ s❡❥❛♠ x /∈ N ❡ y ∈ N✳ ❙❡ (x, y) 6= 1✱ ❡♥tã♦✱ ✈❛❧❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s✿ 2 ∗ y ∗ ∗

  ✭✐✮ y ∈ G ✱ xy = yx ❡ yx = x ✳ 2 2 ∗ ∗ ∗ ∗ , y ∗ y = x y

  ✭✐✐✮ x ∈ G ✱ xy = x ❡ xy ✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ (x, y) 6= 1✱ ♣❡❧♦ ▲❡♠❛ ✹✳✸✱ s❛❜❡♠♦s q✉❡ x, y, xy /∈ G ✱ ❧♦❣♦✱ ♣♦r ❤✐♣ót❡s❡ ∗ ∗ 0 = [x + x , y ] ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ − y

  = xy + x y + y x + y x y , − xy − x − yx − yx

  ♦✉ s❡❥❛ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ xy + x y + y x + y x = xy + x y + yx + yx . ∗ ∗ ∗ y, y x ❈♦♠♦ charR 6= 2 ❡ xy /∈ {x }✱ t❡♠♦s q✉❡ ♦ ♠❡♠❜r♦ ❡sq✉❡r❞♦ ❞❡ss❛ ú❧t✐♠❛ ❡q✉❛çã♦ ∗ ∗ ∗ y , yx

  ♥ã♦ s❡ ❛♥✉❧❛✱ ♣♦rt❛♥t♦ xy ∈ {x }✱ ❥á q✉❡ y / ∈ G ❡ (x, y) 6= 1✳ ∗ ∗ ∗ ∗ y x = yx ❙❡ xy = x ✱ ❡♥tã♦ y ✱ ♣♦rt❛♥t♦ ❛ ❡q✉❛çã♦ s❡ r❡❡s❝r❡✈❡ ♣♦r ∗ ∗ ∗ ∗ ∗ ∗ x y + y x = xy + yx , y

  ❡ ❝♦♠♦ charR 6= 2 ❡ x 6= yx ✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦ (x, y) = 1✱ r❡st❛ ❝♦♠♦ ♣♦ss✐❜✐❧✐❞❛❞❡ ∗ ∗ x y = xy ✳ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 2 2 y = x y y = x yy = x(y ) ∗

  ◆♦t❡ q✉❡ s❡ xy = x ✱ ❡♥tã♦ xy ✱ ♦✉ s❡❥❛ y ∈ G ✳ 2 ❆♥❛❧♦❣❛♠❡♥t❡✱ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ x ∈ G ❡✱ ♣♦rt❛♥t♦✱ ✭✐✐✮ ♦❝♦rr❡✳ ∗ ∗ ∗ ∗ x = xy

  ❙❡ xy = yx ✱ ❡♥tã♦ y ❡ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ s❡ r❡❡s❝r❡✈❡ ♣♦r ∗ ∗ ∗ ∗ x y + y x = yx + x y , y ♣♦rt❛♥t♦✱ ❞❡s❞❡ q✉❡ charR 6= 2 ❡ y /∈ G ✱ t❡♠♦s yx = x ✳

  ❈♦♠♦ (xy, y) 6= 1✱ ❡ xy /∈ N✱ ❛♣❧✐❝❛♥❞♦ ♦ q✉❡ ♣r♦✈❛♠♦s ❛♥t❡r✐♦r♠❡♥t❡ ❛ xy ∗ ∗ ∗ y ❡ y✱ t❡♠♦s q✉❡ (xy)y = y(xy) ♦✉ (xy)y = (xy) ✳ ❙❡ ♦ ♣r✐♠❡✐r♦ ❝❛s♦ ♦❝♦rr❡✱ ❡♥tã♦ 2 ∗ ∗ ∗ ∗ ∗ xy = y yx = y xy x y ∗ ∗ ∗ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ xy = y ✱ ❡ ❞❡s❞❡ q✉❡ yx = x ✱ ❛♣❧✐❝❛♥❞♦ ❛ 2 ∗ ∗ y = y x = xy ∗ y

  ✐♥✈♦❧✉çã♦✱ x ✱ ❛ss✐♠✱ ✭✐✐✮ ♦❝♦rr❡ ❡✱ ♣♦rt❛♥t♦✱ y ∈ G ✳ ❙❡ (xy)y = (xy) ✱ 2 ∗ ∗ ∗ ∗ 2 2 = y x y = x(y ) ∗

  ❡♥tã♦ xy ✱ ♦✉ s❡❥❛✱ y ∈ G ✱ ❧♦❣♦ t❡♠♦s ✭✐✮✳ ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ss❡ ❧❡♠❛✱ ♣♦❞❡♠♦s r❡❞✉③✐r ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ s✉❜✲

  ❣r✉♣♦ N✱ r❡st❛♥❞♦ ❛♣❡♥❛s ❝♦♠♦ ♣♦ss✐❜✐❧✐❞❛❞❡ ❛ ❡str✉t✉r❛ ✉s✉❛❧ ❞❡ q✉❡ N ♣♦ss✉✐ ✉♠ s✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦ ❞❡ ♦r❞❡♠ ♥♦ ♠á①✐♠♦ 2✳

   ✺✺

  ❈♦r♦❧ár✐♦ ✹✳✻✳ ❙❡ RG é ❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ N s❛t✐s❢❛③ ♦ ✐t❡♠ ✭✶✮ ♦✉ ✭✸✮ ❞♦ ❚❡♦r❡♠❛ ✹✳✶✳ ❉❡♠♦♥str❛çã♦✳ ◆♦t❡ q✉❡ s❡ N s❛t✐s❢❛③ ♦ ✐t❡♠ ✭✷✮ ❞♦ t❡♦r❡♠❛✱ ❡♥tã♦ ♦ ▲❡♠❛ ✹✳✸ ✐♠♣❧✐❝❛ q✉❡ G é ❛❜❡❧✐❛♥♦✳ ∗ )

  P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ♦ ✐t❡♠ ✭✹✮ ♦✉ ✭✺✮ ♦❝♦rr❡✱ ❡st❛♠♦s ♥♦ ❝❛s♦ (Z(G) ∩ G \N 6= ∅✱ 2 =

  ❥á q✉❡ R {0} ❝❛s♦ charR = 3✳ ❙❡ ✭✹✮ ♦❝♦rr❡✱ s❡❥❛♠ x, y ∈ N ❡ A ✉♠ s✉❜❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❞❡ í♥❞✐❝❡ 2 ❡♠ N t❛❧ q✉❡ A ⊂ G ✳ ❙❡ x ♦✉ y ∈ A✱ ❡♥tã♦ (x, y) = 1 ❥á q✉❡ A ⊂ G ⊂ Z(G)❀ s❡ x, y / ∈ A✱ ❡♥tã♦✱

  ❝♦♠♦ [N : A] = 2✱ xy ∈ A ⊂ Z(G)✱ ❡ ♣♦rt❛♥t♦ (x, y) = (x, xy) = 1✱ ❛ss✐♠ N é ❛❜❡❧✐❛♥♦ ❡ s❛t✐s❢❛③ ✭✶✮✳ 3 ❙❡ ✭✺✮ ♦❝♦rr❡✱ n ∈ G ∀n ∈ N✱ ♣♦rt❛♥t♦ ♦ ▲❡♠❛ ✹✳✺ ❣❛r❛♥t❡ q✉❡ (N, G\N) = 1✱ 2

  ♣♦✐s s❡ ❡①✐st✐r n ∈ N t❛❧ q✉❡ (n, G\N) 6= {1}✱ ❡♥tã♦✱ ♣❡❧♦ ▲❡♠❛ ✹✳✺✱ n ∈ G ✱ ♦✉ s❡❥❛✱ 3 ∗ ∗ ∗ 2 2 n = n (n ) = n n ✱ ✐♠♣❧✐❝❛♥❞♦ ❡♠ n ∈ G ⊂ Z(G)✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❉❡ss❛ ❢♦r♠❛✱ t♦♠❛♥❞♦ x, y /∈ N✱ ❡♥tã♦ xy ∈ N ❡ ♣♦rt❛♥t♦ (x, y) = (x, xy) = 1✱ ❛ss✐♠ (G\N, G\N) = 1❀

  ♣♦r ✜♠ s❡ x, y ∈ N✱ t♦♠❛♥❞♦ z ∈ Z(G)\N✱ t❡♠♦s q✉❡ zx, zy /∈ N ❡ ♣♦rt❛♥t♦ (x, y) = (zx, zy) = 1

  ❡ ❛ss✐♠✱ G é ❛❜❡❧✐❛♥♦ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ✈❛❧❡ ✭✶✮✳ ▲❡♠❛ ✹✳✼✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❝♦♠✉t❛t✐✈♦ ❡ s❡❥❛♠ x, y /∈ N✳ ❙❡ (x, y) 6= 1✱ ❡♥tã♦✱ ✈❛❧❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s✿ 2 ∗ y / ∗ ∗ ∗ 2

  ✭✐✮ y ∈ G ✱ xy = yx ❡ yx = x ✳ ❆❧é♠ ❞✐ss♦✱ s❡ x ∈ G ✱ ❡♥tã♦ N s❛t✐s❢❛③ ✭✸✮ ❡ y = sy 2 ✱ ❝♦♠ s s❡♥❞♦ ♦ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ ❞❡ N✳ ∗ ∗ 2 x / ✭✐✐✮ x ∈ G ✱ xy = y ❡ yx = xy ✳ ❆❧é♠ ❞✐ss♦✱ s❡ y ∈ G ✱ ❡♥tã♦ N s❛t✐s❢❛③ ✭✸✮ ❡ x = sx

  ✱ ❝♦♠ s s❡♥❞♦ ♦ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ ❞❡ N✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ (x, y) 6= 1✱ ♣❡❧♦ ▲❡♠❛ ✹✳✸✱ t❡♠♦s q✉❡ x, y /∈ G ✳ P♦r ❤✐♣ót❡s❡ t❡♠♦s ∗ ∗ 0 = [x + x , y + y ] ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

  = xy + xy + x y + x y x x , − yx − yx − y − y

  ♦✉ s❡❥❛ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ xy + xy + x y + x y = yx + yx + y x + y x . ∗ , y x, y x x ∗ ∗ ∗ ∗ ∗ ∗ ✭✹✳✶✮ ❈♦♠♦ charR 6= 2 ❡ x, y /∈ G ✱ ❡♥tã♦ xy ∈ {yx }✳ ❆❧é♠ ❞♦ ♠❛✐s✱ s❡ xy = y ✱ ❡♥tã♦ xy ∈ G ✱ ❛ss✐♠✱ ♣❡❧♦ ▲❡♠❛ ✹✳✸✱ (x, xy) = 1✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ (x, y) = 1✱ ✉♠ ∗ ∗

  , y x ❛❜s✉r❞♦✱ ❧♦❣♦ xy ∈ {yx }✳ ∗ ∗ ∗ ∗ x = xy

  ❙❡ xy = yx ✱ ❡♥tã♦ y ❡ ✭✹✳✶✮ s❡ r❡❡s❝r❡✈❡ ♣♦r ∗ ∗ ∗ ∗ x y + x y = yx + y x,

   ✺✻ y ❡ ♣❡❧♦ ♠❡s♠♦ ❛r❣✉♠❡♥t♦ ❛♥t❡r✐♦r✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ yx = x ✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ ∗ ∗ xy = y x

  ✱ ❡♥tã♦ yx = xy ✳ ∗ ∗ y ❙✉♣♦♥❤❛ q✉❡ xy = yx ✱ yx = x ✳ ❈♦♠♦ (yx, y) 6= 1 ❡ yx ∈ N✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r 2

  ♦ ▲❡♠❛ ✹✳✺ ❡ ❛ss✐♠ s❡ ♦ ✐t❡♠ ✭✐✐✮ ❞❡ss❡ ❧❡♠❛ ♦❝♦rr❡r✱ t❡♠♦s q✉❡ y ∗ ❀ s❡ ♦ ✐t❡♠ ✭✐✮ ∗ ∗ ∗ ∗ ∗ ∈ G y = yx ♦❝♦rr❡r✱ ❡♥tã♦ y(yx) = (yx)y ✱ ♦✉ s❡❥❛ yx = xy ✱ ❛ss✐♠✱ ❛♣❧✐❝❛♥❞♦ ❛ ✐♥✈♦❧✉çã♦✱ x ✱ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2

  = x y = y x = x y y = x yy = ♣♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ xy = yx ✱ ❧♦❣♦ xy 2 2 x(y ) ∗

  ✱ ❡ ❝♦♠ ✐ss♦✱ y ∈ G ✳ ∗ ∗ 2 2 2 y = y(x ) ∗ ◆♦t❡ q✉❡ ❞❡ xy = yx ✱ ❝♦♥❝❧✉í♠♦s q✉❡ x ✱ ♦✉ s❡❥❛✱ x ∈ G s❡✱ ❡ 2 2 2 2

  , y) = 1 / , y) , xy) s♦♠❡♥t❡ s❡✱ (x ✱ ♣♦rt❛♥t♦✱ s❡ x ∈ G ✱ (x 6= 1✱ ❧♦❣♦ (x 6= 1✱ ♦✉ s❡❥❛✱ N é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ❡ ♣♦rt❛♥t♦✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✹✳✻✱ t❡♠♦s q✉❡ N s❛t✐s❢❛③ ♦ ✐t❡♠ ✭✸✮ ❞♦ ❚❡♦r❡♠❛ ✹✳✶✳ 2 2

  ) = sx 2 ❉❡ss❛ ❢♦r♠❛ (x ✱ ❝♦♠ s s❡♥❞♦ ♦ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ ❞❡ N✱ ❧♦❣♦ 2 ∗ −2 2 2 2 x y = y(x ) = yx s yx , y) 2 ✱ ♦✉ s❡❥❛ ys = x ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ (x 6= 1✱ ❛♣❧✐❝❛♥❞♦ 2 ∗ ∗ ∗ ∗ ∗ 2 2 2 2 2 2 y = y x y = (x ) y = x sy y = y x ♦ ▲❡♠❛ ✹✳✺ ❛ y ❡ x ✱ ♦❜t❡♠♦s q✉❡ x ♦✉ x ✳ ❙❡ x ✱ ∗ −2 ∗ −1 ∗ 2 2 2 2

  = x yx = sy y = x sy y = y / ❡♥tã♦ y ❀ s❡ x ✱ ❡♥tã♦ sy = s ✳ P♦rt❛♥t♦ s❡ x ∈ G ✱

  = sy ❡♥tã♦ N s❛t✐s❢❛③ ✭✸✮ ❡ y ✱ ❝♦♠ s ♦ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ ❞❡ N✳

  ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♣♦❞❡♠♦s ♣r♦✈❛r ♦ ✐t❡♠ ✭✐✐✮✳ 2 ❖❜s❡r✈❡ q✉❡ ♥♦s ❝❛s♦s tr❛t❛❞♦s ♥♦s ❈❛♣ít✉❧♦s ✷ ❡ ✸✱ ♣✉❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ x

  ∈ G ∗ ,

  ∀x ∈ G✱ ❡♥tr❡t❛♥t♦ ♥ã♦ ♣♦ss✉í♠♦s ❢❡rr❛♠❡♥t❛s s✉✜❝✐❡♥t❡s ♣❛r❛ ❣❛r❛♥t✐r ❡st❛ ♣r♦♣r✐✲ ❡❞❛❞❡ s♦❜ ❛ ❤✐♣ót❡s❡ ❞❡ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❛♥t✐ss✐♠étr✐❝♦s✳ ❉❡st❛ ❢♦r♠❛✱ ✐r❡♠♦s ❞✐✈✐❞✐r ♥♦ss♦ ❡st✉❞♦ ❛❞♠✐t✐♥❞♦ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ∗ ♣♦ss✉✐r ♦✉ ♥ã♦ ❡ss❛ ❝❛r❛❝t❡ríst✐❝❛✳

  2 ✹✳✶✳✶ G 6⊂ G

  ∗ 2 ◆❡st❛ s❡çã♦ ✐r❡♠♦s ❛❞♠✐t✐r q✉❡ ❡①✐st❡ ❛♦ ♠❡♥♦s ✉♠ ❡❧❡♠❡♥t♦ x ∈ G t❛❧ q✉❡ x /

  ∈ G ✳ ❖ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦ ❛✜r♠❛ q✉❡✱ ♥❡st❛s ❝♦♥❞✐çõ❡s✱ s❡ G é ♥ã♦ ❛❜❡❧✐❛♥♦ ❡♥tã♦ N s❛t✐s❢❛③ ♦ ✐t❡♠ ✭✸✮ ❞♦ ❚❡♦r❡♠❛ ✹✳✶✳ 2 ▲❡♠❛ ✹✳✽✳ ❙❡ RG é ❝♦♠✉t❛t✐✈♦✱ N é ❛❜❡❧✐❛♥♦ ❡ G é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ x ∈ G ∀x ∈ G✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ x /∈ G ∗ ✳

  ❙✉♣♦♥❤❛ q✉❡ x /∈ Z(G)✳ ❙❡ x ∈ N✱ ❝♦♠♦ N é ❛❜❡❧✐❛♥♦✱ ❡①✐st❡ y /∈ N t❛❧ q✉❡ 2 (x, y)

  6= 1✱ ♣♦rt❛♥t♦ ♦ ▲❡♠❛ ✹✳✺ ❣❛r❛♥t❡ q✉❡ x ∈ G ∗ ✳ ❙✉♣♦♥❤❛ q✉❡ x /∈ N✳ ◆❡ss❡ ❝❛s♦✱ s❡❥❛ y ∈ G t❛❧ q✉❡ (x, y) 6= 1 ❡ ♥♦t❡ q✉❡ y /∈ G ⊂ Z(G)✳ ❙❡ y / ∈ N✱ ❝♦♠♦ N é 2

  ❛❜❡❧✐❛♥♦✱ ♦ ▲❡♠❛ ✹✳✼ ❛✜r♠❛ q✉❡ x ∈ G ❀ s❡ y ∈ N✱ ❡♥tã♦ xy /∈ N✱ ❛❧é♠ ❞✐ss♦✱ ❞❡s❞❡ q✉❡ (x, y) 6= 1✱ ❡♥tã♦ (x, xy) 6= 1✱ ♣♦rt❛♥t♦✱ ♣❡❧♦ ▲❡♠❛ ✹✳✸✱ s❛❜❡♠♦s q✉❡ xy /∈ G ✱ ❛ss✐♠✱ 2 ❛♣❧✐❝❛♥❞♦ ♥♦✈❛♠❡♥t❡ ♦ ▲❡♠❛ ✹✳✼ ❛ x ❡ xy✱ ❡♥❝♦♥tr❛♠♦s q✉❡ x ∈ G ✳

  ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ x ∈ Z(G)✳ ❈♦♠♦ G é ♥ã♦ ❛❜❡❧✐❛♥♦✱ t♦♠❡ y /∈ Z(G)✱ ❧♦❣♦ 2 2 2 2 2 y / , y y = (xy) =

  ∈ G ❡ xy /∈ Z(G)✱ ♣♦rt❛♥t♦ ♣❡❧♦ ♣r♦✈❛❞♦ ❛❝✐♠❛✱ (xy) ∈ G ✱ ❛ss✐♠ x

   2 ✺✼ /

  ❈♦r♦❧ár✐♦ ✹✳✾✳ ❙❡ RG é ❝♦♠✉t❛t✐✈♦✱ G é ♥ã♦ ❛❜❡❧✐❛♥♦ ❡ ❡①✐st❡ x ∈ G t❛❧ q✉❡ x ∈ G ✱ ❡♥tã♦ N ✈❡r✐✜❝❛ ♦ ✐t❡♠ ✭✸✮ ❞♦ ❚❡♦r❡♠❛ ✹✳✶✳ 2 Pr♦♣♦s✐çã♦ ✹✳✶✵✳ ❙❡ RG é ❝♦♠✉t❛t✐✈♦✱ R ∗ 2 6= {0} ♦✉ (G ∩ Z(G))\N 6= ∅✱ G é ♥ã♦ 2

  / / ❛❜❡❧✐❛♥♦ ❡ ❡①✐st❡ x ∈ G t❛❧ q✉❡ x ∗ ✱ ❡♥tã♦ A = hx ∈ G : x ∗ ∗ = x , ∈ G ∈ G i é ✉♠ s✉❜❣r✉♣♦ ∗ y ❛❜❡❧✐❛♥♦ ❞❡ í♥❞✐❝❡ 2 ❡♠ G t❛❧ q✉❡ G ⊂ A ❡ x ∀x ∈ A ❡ y / ∈ A✳ ∗ , y / ∗ ∗ 2 2

  ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ x, y ∈ G t❛✐s q✉❡ x ∈ G ✳ ◆❛t✉r❛❧♠❡♥t❡✱ x, y /∈ G ✳ ❈♦♠♦ N ✈❡r✐✜❝❛ ♦ ✐t❡♠ ✭✸✮ ❞♦ ❚❡♦r❡♠❛ ✹✳✶✱ t❡♠♦s q✉❡ x, y /∈ N✱ ❛ss✐♠✱ ♦ ▲❡♠❛ ✹✳✼ ❣❛r❛♥t❡ q✉❡ (x, y) = 1

  ✱ ♣♦rt❛♥t♦ A é ❛❜❡❧✐❛♥♦✳ 2 2 / ∗ ∗ ∗ /

  ❙❡❥❛ x ∈ G t❛❧ q✉❡ x ∈ G ❡ y ∈ G ⊂ Z(G)✳ ◆♦t❡ q✉❡ (xy) ∈ G ✱ ♣♦✐s ❝❛s♦ 2 2 2 2 ∗ ∗ ∗ ∗ 2 2 2 2 2 y = (xy) = ((xy) ) = (x ) (y ) = (x ) y ❝♦♥trár✐♦✱ x

  ✱ ♦ q✉❡ ✐♠♣❧✐❝❛r✐❛ ❡♠ x ∈ G ✱ ✉♠ ❛❜s✉r❞♦✱ ❧♦❣♦ xy ∈ A ❡✱ ❛ss✐♠ y ∈ A✱ ❥á q✉❡ x ∈ A✱ ♣♦rt❛♥t♦ G ⊂ A✳ y

  = x , = y P❛r❛ ✈❡r✐❝❛r q✉❡ x ∀x ∈ A ❡ y / ∈ A✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ x 2 x , /

  ∀x ∈ A t❛❧ q✉❡ x ∈ G ✳ ❈♦♠♦ x /∈ N✱ ♦s ▲❡♠❛s ✹✳✺ ❡ ✹✳✼ ❣❛r❛♥t❡♠ q✉❡ xy ∈ 2 {yx, yx }✳ ❙❡ xy = yx✱ ❝♦♠♦ x ∈ A ❡ y / ∈ A✱ ❡♥tã♦ xy / ∈ A✱ ❧♦❣♦✱ (xy) ∈ G ✱ ❛ss✐♠ 2 2 2 2 ∗ ∗ ∗ ∗ 2 2 2 2 2 x y = (xy) = ((xy) ) = (x ) (y ) = (x ) y ∗ ∗ y ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ x ∈ G ✱ ✉♠ ❛❜s✉r❞♦✱

  = x ♣♦rt❛♥t♦ xy = yx ✱ ♦✉ s❡❥❛✱ x ✳

  P♦r ✜♠✱ ✈❛♠♦s ♠♦str❛r q✉❡ (G : A) = 2✳ P❛r❛ ✐ss♦✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ xy ∈ A, ∀x, y / ∈ A✳ ❙❡❥❛♠ a ∈ A ❡ x, y / ∈ A ❡ s✉♣♦♥❤❛ q✉❡ xy / ∈ A✳ P❡❧♦ ♣r♦✈❛❞♦ ❛❝✐♠❛ ∗ ∗ xy x y ∗ y ∗ ∗

  = a = (a ) = (a ) = (a ) = a ❡ ♦ ❢❛t♦ ❞❡ a ∈ A✱ t❡♠♦s q✉❡ a ✱ ✉♠ ❛❜s✉r❞♦✱ ❥á q✉❡ a / ∗

  ∈ G ✱ ❧♦❣♦ (G : A) = 2✳ ❱❛♠♦s ❛♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡ss❛ s❡çã♦✳ 2

  ❚❡♦r❡♠❛ ✹✳✶✶✳ ❙✉♣♦♥❤❛ q✉❡ G é ♥ã♦ ❛❜❡❧✐❛♥♦ ❡ R 6= {0} ♦✉ (G ∩ Z(G))\N 6= ∅✳ ❙❡ 2 /

  ❡①✐st❡ x ∈ G t❛❧ q✉❡ x ∈ G ✱ ❡♥tã♦ RG é ❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ charR = 4 ❡ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ✈❡r❞❛❞❡✐r❛s✿ 2 /

  ✭✐✮ A = hx ∈ G : x ∈ G i é ❛❜❡❧✐❛♥♦ ❡ (G : A) = 2✳ ′ ∗ = ✭✐✐✮ G ⊂ Z(G)✱ N {1, s} ❡ n ∈ {n, sn} , ∀n ∈ N✳

  ✭✐✐✐✮ ❆ ✐♥✈♦❧✉çã♦ ∗ é ❞❛ ❢♦r♠❛ ( sx , s❡ x /∈ A x = −1 y xy , s❡ x ∈ A,

  ♣❛r❛ q✉❛❧q✉❡r y /∈ A✳ ✭✐✈✮ (x, y) ∈ {1, s} , ∀x, y /∈ A ∪ N✳

  ❊♠ ♣❛rt✐❝✉❧❛r Z(G) ⊂ A✳

   ✺✽

  ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❝♦♠✉t❛t✐✈♦✳ ❆ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ❣❛r❛♥t❡ ✭✐✮✳ ❉♦ ▲❡♠❛ ✹✳✸ ❡ ♦ ❈♦r♦❧ár✐♦ ✹✳✾✱ ❝♦♥❝❧✉í♠♦s q✉❡ charR = 4

  ✱ G ∗ ⊂ Z(G) ❡ N ✈❡r✐✜❝❛ ❛s ❝♦♥❞✐çõ❡s ❞♦ ✐t❡♠ ✭✐✐✮✳

  ❱❛♠♦s ♠♦str❛r q✉❡ ✭✐✐✐✮ ♦❝♦rr❡✳ Pr✐♠❡✐r♦ ✈❛♠♦s ♠♦str❛r q✉❡ N\(G ∗ ∪ A) 6= ∅✳ ❈♦♠♦ (G : A) = 2 = (G : N)✱ 2

  ❡♥tã♦ N 6⊂ A ♦✉ N = A✱ ♣♦ré♠ ♦ s❡❣✉♥❞♦ ❝❛s♦ ♥ã♦ ♦❝♦rr❡ ❞❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ q✉❡ n ∈ G ✳ ∗ ∗ / ∗ 2 P♦r ♦✉tr♦ ❧❛❞♦✱ (N\A) 6⊂ G ✱ ♣♦✐s s❡ n ∈ N \A✱ t♦♠❛♥❞♦ x ∈ A t❛❧ q✉❡ x ∈ G ✱ 2 2 s❛❜❡♠♦s q✉❡✱ ♣♦r ✭✐✐✮✱ x /∈ N ❡ ❛❧é♠ ❞✐ss♦ x ∈ N✱ ❞❡ss❛ ❢♦r♠❛✱ nx ∈ N✱ ♣♦ré♠ 2 ∗ ∗ ∗ ∗ ∗ 2 2 2 2 2 2 (nx ) = (x ) n = (x ) n = n(x ) / ∗ ∗ 6= nx ✱ ❥á q✉❡ x ∈ G ✱ ♣♦rt❛♥t♦ nx ∈ N\(G ∪ A)✳

  ❙❡❥❛ x ∈ N\(G ∪ A) ❡ y ∈ G\A✳ ❉❡s❞❡ q✉❡ (G : A) = 2✱ ♣♦❞❡♠♦s t♦♠❛r ∗ −1 ∗ ∗ ∗ y = ax = x ax = x (a) = −1 ❝♦♠ a ∈ A✳ ◆♦t❡ q✉❡✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✹✳✶✵✱ a ✱ ❛ss✐♠ y

  (sx)(x ax) = sax = sy ✳ ❊ss❡ ❢❛t♦ ❡ ❛ Pr♦♣♦s✐çã♦ ✹✳✾ ❣❛r❛♥t❡♠ q✉❡ ❛ ✐♥✈♦❧✉çã♦ ♣♦ss✉✐ ❛

  ❢♦r♠❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r ✭✐✐✐✮✳ P❛r❛ ✈❡r✐✜❝❛r ✭✐✈✮✱ ♥♦t❡ q✉❡ ♦ ▲❡♠❛ ✹✳✼ ❛✜r♠❛ q✉❡ s❡ x, y /∈ N sã♦ t❛✐s q✉❡ ∗ ∗

  (x, y) , y x 6= 1✱ ❡♥tã♦ xy ∈ {yx }✱ ❡ ♣♦r ✭✐✐✐✮✱ ✐ss♦ ❡q✉✐✈❛❧❡ ❛ xy = syx✱ ❥á q✉❡ s ⊂ Z(G)✳

  P❛r❛ ❛ r❡❝í♣r♦❝❛✱ s✉♣♦♥❤❛ q✉❡ charR = 4 ❡ ♦s ✐t❡♥s ✭✐✮✲✭✐✈✮ s❡❥❛♠ ✈❡r✐✜❝❛❞♦s✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♥♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s✱ ♣❛r❛ ✈❡r✐✜❝❛r q✉❡ RG é ❝♦♠✉t❛✲ t✐✈♦✱ é s✉✜❝✐❡♥t❡ ✈❡r✐✜❝❛r q✉❡ ♦s ❣❡r❛❞♦r❡s ❞❡ RG ❝♦♠✉t❛♠ ❡♥tr❡ s✐✳ ❉❡✈✐❞♦ ❛ ❡str✉t✉r❛

  ❞♦s ❣❡r❛❞♦r❡s ❡ ♦ ❢❛t♦ ❞❡ G ∗ ⊂ Z(G)✱ é s✉✜❝✐❡♥t❡ ✈❡r✐✜❝❛r q✉❡ (x − σ(x)) ❝♦♠✉t❛ ❝♦♠ (y ∗

  − σ(y)) ❝♦♠ x, y / ∈ G t❛✐s q✉❡ (x, y) 6= 1✳ ❈♦♠♦ A é ❛❜❡❧✐❛♥♦✱ ❞❡✈❡♠♦s ❛♥❛❧✐s❛r ❛♣❡♥❛s q✉❛♥❞♦ x ♦✉ y /∈ A✱ ❛❧é♠ ❞✐ss♦✱ s❡ x, y ∈ N✱ ♦ ✐t❡♠ ✭✐✐✐✮ ❞♦ ❚❡♦r❡♠❛ ✹✳✶ ❣❛r❛♥t❡ ♦ r❡s✉❧t❛❞♦✳ ❘❡st❛✱ ♣♦rt❛♥t♦✱ ✈❡r✐✜❝❛r q✉❛♥❞♦ x ♦✉ y /∈ N✳ ∗ ∗ y

  = x = sy = ys 2 ❙✉♣♦♥❤❛ q✉❡ x ∈ A\N ❡ y /∈ A✱ ❛ss✐♠ x ❡ y ✱ ♣♦rt❛♥t♦✱ ❝♦♠♦ y , s ∈ G ⊂ Z(G)✱ t❡♠♦s ∗ ∗ y

  (x + x )(y ) = (x + x )(y − σ(y)y − σ(y)ys) y y

  = xy + x y ys y −1 y −1 − σ(y)xys − σ(y)x 2 2 = yx + y xy s xy s y y − σ(y)yx − σ(y)y = yx + yx

  − σ(y)ysx − σ(y)ysx y = (y )

  − σ(y)ys)(x + x ∗ ∗ = (y )(x + x ). − σ(y)y ∗ ∗

  ) ) ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡ (x − σ(x)x ❝♦♠✉t❛ ❝♦♠ (y − σ(y)y s❡ x /∈ A ❡ y ∈ A\N✳ ∗ ∗

  = sx = sy ❙✉♣♦♥❤❛ q✉❡ x /∈ N ∪ A ❡ y ∈ N✱ ❛ss✐♠ x ❡ y ✱ ♣♦rt❛♥t♦✱ ❝♦♠♦

  2 ✺✾ s = 1

  ∈ Z(G) ❡ s ✱ t❡♠♦s ∗ ∗ (x + x )(y ) = (x + xs)(y

  − y − ys) = xy + xsy

  − xys − xsys = 0 = yx + yxs

  − ysx − ysxs = (y

  − ys)(x + xs) ∗ ∗ = (y )(x + x ). − y ∗ ∗

  ) ) ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ (x − σ(x)x ❝♦♠ (y − σ(x)y ✱ ❝❛s♦ x ∈ N ❡ y /∈ N ∪ A✳

  P♦r ✜♠ r❡st❛ ✈❡r✐✜❝❛r ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❝❛s♦ x, y /∈ N ∪ A✳ ◆❡ss❡ ❝❛s♦✱ ♣♦r ✭✐✈✮✱ ∗ −1 ∗ −1 x = y y t❡♠♦s q✉❡ (x, y) = s = x ✱ ♣♦rt❛♥t♦ xy = yx(x, y) = yxs✱ ❧♦❣♦ ∗ ∗

  (x + x )(y + y ) = (x + xs)(y + ys) = xy + xsy + xys + xsys 2 2

  = yxs + yxs + yxs + ysx = yxs + yx + ysxs + ysx = (y + ys)(x + xs) ∗ ∗ = (y + y )(x + x ),

  ❝♦♥❝❧✉✐♥❞♦ ❛ r❡❝í♣r♦❝❛✳ P❛r❛ ✈❡r✐✜❝❛r q✉❡ Z(G) ⊂ A✱ ❜❛st❛ ♥♦t❛r q✉❡ s❡ ❡①✐st✐r y ∈ Z(G)\A✱ ❡♥tã♦✱ ♣❛r❛ y 2

  = x = x q✉❛❧q✉❡r x ∈ A✱ x ❡ ♣♦rt❛♥t♦ x ∈ G ∀x ∈ A✱ ❝♦♥tr❛❞✐çã♦✳

  2 ✹✳✶✳✷ G ⊂ G ∗

  ◆❡st❛ s❡çã♦ ✐r❡♠♦s ❝♦♠♣❧❡t❛r ♥♦ss♦ ♣❛rt✐❝✉❧❛r ❡st✉❞♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❛♥t✐ss✐♠étr✐❝♦s✱ ❛❞♠✐t✐♥❞♦ ❛❣♦r❛ q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ G ❡❧❡✈❛❞♦ ❛ s❡❣✉♥❞❛ ♣♦tê♥❝✐❛ é s✐♠étr✐❝♦✳ 2 =

  Pr♦♣♦s✐çã♦ ✹✳✶✷✳ ❙✉♣♦♥❤❛ RG é ❝♦♠✉t❛t✐✈♦✳ ❙❡ x ∈ G ∀x ∈ G✱ ❡♥tã♦ x 2 c x x, x = 1 ∗ x ∀x ∈ G✱ ♦♥❞❡ c ∈ G ❡ c ✳ ❆❧é♠ ❞✐ss♦✱ G é ✉♠ 2✲❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❡❧❡♠❡♥✲ xy = c x c y (x, y) xy x , c y , (x, y) t❛r✱ c ❡ s❡ (x, y) 6= 1✱ ❡♥tã♦ c ∈ {c }✳

  ❉❡♠♦♥str❛çã♦✳ ❆✜r♠❛çã♦✿ (x, y) ∈ G ∀x, y ∈ G✳ ❙❡❥❛♠ x, y ∈ G t❛✐s q✉❡ (x, y) 6= 1✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ ♥❛❞❛ t❡♠♦s ❛ ♣r♦✈❛r✳

  ◆♦t❡ q✉❡✱ ♣❡❧♦ ▲❡♠❛ ✹✳✸✱ x, y, xy /∈ G ✳ ❙❡ x, y ∈ N✱ s❡❣✉❡ ❞♦ ❈♦r♦❧ár✐♦ ✹✳✻ q✉❡ (x, y) ∈ G ✳ ∗ −1 −1 −1 ∗ 2 y xy = x x

  ❖❜s❡r✈❡ q✉❡ s❡ xy = yx ✱ ❡♥tã♦ (x, y) = x ∈ G ✱ ❥á q✉❡ x ∈ −1 ∗ G x ∗ ∗

  ✱ ✐♠♣❧✐❝❛ ❡♠ x ∈ G ✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ s❡ yx = xy ✱ ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡

  ✻✵ x / ∈ N ❡ y ∈ N✳ ❆♣❧✐❝❛♥❞♦ ♦ ▲❡♠❛ ✹✳✺✱ ❝❛s♦ ♦ ✐t❡♠ ✭✐✮ ♦❝♦rr❛✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ♦ ♠❡s♠♦ ∗ ∗ ∗ ∗ y = x y

  ❛r❣✉♠❡♥t♦ ❛❝✐♠❛✱ ♣♦rt❛♥t♦✱ s✉♣♦♥❤❛ q✉❡ xy = x ❡ xy ✳ 2 2 2 −1 −1 −1 −1 ∗ −∗ −1 y = yx y x = xy x = x y x = ❈♦♠♦ x ∗ ✱ ♣♦rt❛♥t♦ x ∗ −1 −∗ −1 −1 −1 −1 ∈ G ⊂ Z(G)✱ x x y x y x y x, y) = 1

  ✱ ♦✉ s❡❥❛ x ∗ ✳ ❙❡ (x ✱ t❡♠♦s q✉❡ ∈ N\G −1 −1

  (x, y) = x y xy −1 −1 = yx y x −1 −1 = yxy x ∗ ∗ −∗ −∗ = y x y x −1 −1 = (x, y) . −1 −1 y x, y) y x, y ∗

  ❙❡ (x 6= 1✱ ❡♥tã♦ N s❛t✐s❢❛③ ✭✸✮ ❞♦ ❚❡♦r❡♠❛ ✹✳✶✱ ❥á q✉❡ x ∈ N\G ✳ ◆❡ss❡ ❝❛s♦✱ t♦♠❛♥❞♦ s ❝♦♠♦ ♦ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ ❞❡ N✱ t❡♠♦s −1 −1

  (x, y) = x y xy −1 −1 = yx y xs −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 −1 = yxy x s, 2 2 y xy)(xyx y ) = x y (xy) x y = x y x (xy) y = 1

  ♦✉ s❡❥❛✱ s = (x 2 ✱ ❥á q✉❡✱ (xy) ∈ G ⊂ Z(G)✱ ♣♦rt❛♥t♦ s = 1✱ ✉♠ ❛❜s✉r❞♦✳ ❈♦♥❝❧✉✐♥❞♦ ❛ ❛✜r♠❛çã♦✳

  ❆✜r♠❛çã♦✿ G é ✉♠ 2✲❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❡❧❡♠❡♥t❛r✳ −1 −1 2 , y ) = 1,

  P❛r❛ ♣r♦✈❛r ❡ss❛ ❛✜r♠❛çã♦✱ é s✉✜❝✐❡♥t❡ ✈❡r✐✜❝❛r q✉❡ (x ∀x, y ∈ G✱ −1 −1 −1 −1 y = yxy x ♦✉ s❡❥❛ xyx ✳

  ❙❡❥❛♠ x, y ∈ G✳ ❙❡ (x, y) = 1✱ ♥❛❞❛ t❡♠♦s ❛ ♣r♦✈❛r✳ ❙✉♣♦♥❤❛ q✉❡ (x, y) 6= 1✳ P❡❧♦ ❈♦r♦❧ár✐♦ ✹✳✻✱ s❛❜❡♠♦s N ✈❡r✐✜❝❛ ✭✶✮ ♦✉ ✭✸✮ ❞♦ ❚❡♦r❡♠❛ ✹✳✶✱ ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ s❡ x, y ∈ N✱ ♦ ❢❛t♦ ❛ s❡r ♣r♦✈❛❞♦ ♦❝♦rr❡❀ ♣♦r ♦✉tr♦ ❧❛❞♦✱ s❡ x ♦✉ y /∈ N✱ ❛♣❧✐❝❛♥❞♦ ♦s ∗ ∗ ∗ ∗

  , y x, x y ▲❡♠❛s ✹✳✺ ❡ ✹✳✼✱ s❛❜❡♠♦s q✉❡ xy ∈ {yx }✳ y

  ❙❡ xy = yx ✱ ♣❡❧♦s ♠❡s♠♦s r❡s✉❧t❛❞♦s ❝✐t❛❞♦s ❛❝✐♠❛✱ t❡♠♦s yx = x ✱ ♣♦rt❛♥t♦ −1 −1 −∗ −1 xyx y = xx yy −∗ = xx ∗ −1 ∗ 2 2

  = x x (x = (x ) ) ∗ −1 −1 = x yy x −1 −1 = yxy x . x

  ❖ ❝❛s♦ xy = y é ❛♥á❧♦❣♦✳

  ∗ ∗ −1 −1 ✻✶ y , y )

  ❙❡ xy = x ✱ ❝♦♠♦ (x ∈ G ✱ ❡♥tã♦ −1 −1 −∗ −∗ ∗ ∗ xyx y = y x y x −1 −1 = y x yx −1 −1 −1 −1 = (yy )y (xx )x yx −2 −2 = yy xx yx −1 −1 2 = yxy x , 2

  , y ♦♥❞❡ ❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡ ❞❡ x

  ∈ Z(G)✳ ■ss♦ ❝♦♥❝❧✉✐ ❛ ❛✜r♠❛çã♦✳ ❱❛♠♦s ❛❣♦r❛ à ♣r♦✈❛ ❞❛ Pr♦♣♦s✐çã♦✳ x = x x ∗ −1 ❉❡✜♥✐♥❞♦ c ✱ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ ▲❡♠❛ ✸✳✶✷✱ ♣♦❞❡♠♦s ♠♦str❛r q✉❡ c x

  ∈ Z(G) ∀x ∈ G✳ x = x x = xx = x x = c x ∗ −1 −1 −∗ ∗ −1 2 ❉❛❞♦ x ∈ G✱ t♦♠❡ c ❡ ♥♦t❡ q✉❡ c x ✱ ❥á q✉❡ x ∈ 2

  G = 1 x = c xy xy ∗ ∗ ✱ ♣♦rt❛♥t♦ c x ❡ c ∈ G ✳ ❚♦♠❡ y ∈ G✱ ❛ss✐♠ (xy) ❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❧♦ ∗ ∗ ∗ −1 −1

  = y x = c y yc x x = c x c y yx xy = c x c y (y , x ) = c x c y (x, y) ▲❡♠❛ ✸✳✶✷✱ (xy) ✱ ♦✉ s❡❥❛✱ c ✳

  P♦r ✜♠✱ s❡ (x, y) 6= 1✱ ♦s ▲❡♠❛s ✹✳✺ ❡ ✹✳✼ ❡ ♦ ❈♦r♦❧ár✐♦ ✹✳✻✱ ❣❛r❛♥t❡♠ q✉❡ ✈❛❧❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s✿ ∗ −1 −1

  = y xy = yxy ✭❛✮ x ❀ ∗ −1 −1

  = x yx = xyx ✭❜✮ y ❀ ∗ ∗ ∗ ∗ y = x y

  ✭❝✮ xy = x ❡ xy ✳ x = x x = x y xy = (x, y) xy = (x, y)c y (x, y) = c y −1 ∗ −1 −1 ❆ss✐♠ s❡ ✭❛✮ ♦❝♦rr❡✱ ❡♥tã♦ c ❡ c ❀ y = (x, y) x = c xy ❛♥❛❧♦❣❛♠❡♥t❡✱ s❡ ✭❜✮ ♦❝♦rr❡✱ ❡♥tã♦ c ❡ c ❀ ♣♦r ✜♠✱ s❡ ✭❝✮ ♦❝♦rr❡✱ ❡♥tã♦ ∗ −1 ∗ −1 c x = x x = y y = c y xy = (x, y)

  ❡ c ✱ ❝♦♥❝❧✉✐♥❞♦ ♦ ❧❡♠❛✳ ❆ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦ ♥♦s ❢♦r♥❡❝❡ ✉♠❛ ❡str✉t✉r❛ ❜❛st❛♥t❡ ♣r❡❝✐s❛ ❞❡ G ❡ ❡①✐❜❡

  ✉♠❛ ❢♦rt❡ r❡❧❛çã♦ ❡♥tr❡ ❡ss❡ s✉❜❣r✉♣♦ ❡ ❛ ✐♥✈♦❧✉çã♦ ∗✳ 2 , Pr♦♣♦s✐çã♦ ✹✳✶✸✳ ❙✉♣♦♥❤❛ q✉❡ RG é ❝♦♠✉t❛t✐✈♦✳ ❙❡ x ∈ G ∀x ∈ G ❡ N é ♥ã♦ x , c y ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ (x, y) = 1, ∀x, y t❛✐s q✉❡ c 6= s ❡ ♦❝♦rr❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s✿

  ✭✐✮ G é ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ∗✳ 2 2 = x = t y = u

  ✭✐✐✮ G {1, s, t, u} ≃ C × C ✱ ❡ ❡①✐st❡♠ x, y ∈ G t❛✐s q✉❡ c ❡ c ✳ 2 2 = x

  ✭✐✐✐✮ G {1, s, t, u} ≃ C × C ✱ N é ✉♠ ❙▲❈✲❣r✉♣♦✱ G ⊂ N✱ c ∈ {1, s, t} , ∀x ∈ G ❡ y = t ❡①✐st❡ y ∈ G t❛❧ q✉❡ c ✳

  ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ N é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✹✳✻✱ s❛❜❡♠♦s q✉❡ N s❛t✐s❢❛③ ♦ n ✐t❡♠ ✭✸✮ ❞♦ ❚❡♦r❡♠❛ ✹✳✶❀ ❞❡ss❛ ❢♦r♠❛✱ c ∈ {1, s} , ∀n ∈ N✳

  x / ✻✷

  ❆✜r♠❛çã♦✿ ❙❡ c ∈ {1, s} ♣❛r❛ ❛❧❣✉♠ x / ∈ N✱ ❡♥tã♦ ❡①✐st❡ t ∈ G t❛❧ q✉❡ 2 2 c x ∈ {1, s, t, st = u} , ∀x ∈ G✳ ❆❧é♠ ❞✐ss♦✱ G ≃ C × C ✳ x

  ❋✐①❡ x ∈ G\G ∗ t❛❧ q✉❡ c 6= s✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ♣♦❞❡♠♦s t♦♠❛r x / z /

  ∈ Z(G)✱ ♣♦✐s s❡ z ∈ Z(G) é t❛❧ q✉❡ c ∈ {1, s}✱ ❡♥tã♦✱ ❝♦♠♦ N é ♥ã♦ ❛❜❡❧✐❛♥♦✱ t♦♠❛♥❞♦ n / n = s zn = c z c n (z, n) = c z s zn / ∈ Z(G)✱ t❡♠♦s q✉❡ c ✱ ❛ss✐♠ c ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ c ∈ {1, s} x = t

  ❡ zn /∈ Z(G) ∪ N✳ ❉❡♥♦t❡ ❞❡ss❛ ❢♦r♠❛✱ c ✳ xy x , c y ∗ xy = s y = s x = t ❙❡❥❛ y /∈ G ∪ N✳ ❙❡ (x, y) 6= 1✱ ❡♥tã♦ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✹✳✶✷ ❡ ♦ ▲❡♠❛ ✹✳✼✱ t❡♠♦s q✉❡ c ∈ {c }✱ ❛❞❡♠❛✐s✱ xy ∈ N\G ✱ ❧♦❣♦ c ✱ ❛ss✐♠✱ c ✱ ❥á q✉❡ c ❀ ♦❜s❡r✈❡ xy x y y = c c (x, y) = tc q✉❡ ❛ ♠❡s♠❛ ♣r♦♣♦s✐çã♦ ❣❛r❛♥t❡ q✉❡ (x, y) = t✳ ❙❡ (x, y) = 1✱ ❡♥tã♦ c ✱ y xy xy = c x c y (x, y) ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ c ∈ {t, st}✱ ❥á q✉❡ c ∈ {1, s}✳ ❉❛ ❢ór♠✉❧❛ c ❡ ♦ ❢❛t♦ ❞❡ q✉❡ G é ✉♠ 2✲❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❡❧❡♠❡♥t❛r✱ ❝♦♥❝❧✉í♠♦s q✉❡ G ⊂ {1, s, t, u}✳

  = 2 2 P❛r❛ ♠♦str❛r q✉❡ G {1, s, t, u} ≃ C × C ✱ ❜❛st❛ ♥♦t❛r q✉❡ (x, G\N) 6= {1}✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦✱ t♦♠❛♥❞♦ n ∈ N✱ ❡♥tã♦ xn /∈ N✱ ❧♦❣♦ (x, n) = (x, xn) = 1 ❡ ♥❡st❡ ❝❛s♦ x ∈ Z(G)✱ ❝♦♥tr❛❞✐çã♦✳ P♦rt❛♥t♦ ❡①✐st❡ y /∈ G ∪ N t❛❧ q✉❡ (x, y) 6= 1✱ ❡ ♥❡st❡ ❝❛s♦✱ ❥á ♠♦str❛♠♦s✱ ♣❡❧♦ ♣r✐♠❡✐r♦ ❝❛s♦ ❞♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✱ q✉❡ (x, y) = t✳ ■ss♦ ❝♦♥❝❧✉✐ ❛ ❛✜r♠❛çã♦✳

  ❱❛♠♦s ❛♥❛❧✐s❛r q✉❛tr♦ ❝❛s♦s ❡ ✈❡r✐✜❝❛r q✉❡ ✭✐✮✱ ✭✐✐✮ ♦✉ ✭✐✐✐✮ ♦❝♦rr❡✳ ❈❛s♦ ✶✿ ∅ 6= Z(G)\N ⊂ G ∗ ✳ xn = c x c n (x, n) xn , c x n = 1 ❈♦♠♦ Z(G) 6⊂ N✱ ✜①❡ x ∈ Z(G)\N✳ ❙❡ n ∈ Z(G) ∩ N✱ ♣❡❧♦ ▲❡♠❛ ✹✳✶✷✱ t❡♠♦s q✉❡ c ✱ ❡ ♣♦r ❤✐♣ót❡s❡✱ c ❡ (x, n) = 1✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ c ✱ ♦✉ s❡❥❛✱ ∗ ∗ ∗ ∗ =

  Z(G) = G ✱ ❥á q✉❡ ♣❡❧♦ ▲❡♠❛ ✹✳✸ t❡♠♦s G ⊂ Z(G)❀ s❡ y / ∈ G ∪ N✱ ❝♦♠♦ G Z(G)✱ ∗ xy = c x c y (x, y) = c y ❡♥tã♦ xy ∈ N\Z(G) = N\G ✱ ♦✉ s❡❥❛✱ s = c ✱ ❝♦♥❝❧✉✐♥❞♦ q✉❡ G é ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ∗✱ ♦✉ s❡❥❛✱ ✈❛❧❡ ✭✐✮✳

  ❈❛s♦ ✷✿ ∅ 6= Z(G)\N 6⊂ G ✳ x P♦r ❤✐♣ót❡s❡ ♣♦❞❡♠♦s t♦♠❛r x ∈ Z(G)\N t❛❧ q✉❡ c 6= 1✳ ❈♦♠♦ N é ♥ã♦ xn

  ❛❜❡❧✐❛♥♦✱ t♦♠❡ n ∈ N t❛❧ q✉❡ n /∈ Z(G) ⊂ G ✳ P❡❧❛ ❛✜r♠❛çã♦✱ c ∈ {1, s, t, u}✳ ❈♦♠♦ xn / xn xn = c x c n (x, n) = c x s x ∈ Z(G)✱ t❡♠♦s c 6= 1✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ c ❀ ♥♦t❡ q✉❡ c 6= s✱ ♣♦✐s xn = 1 x = t

  ❝❛s♦ ❝♦♥trár✐♦ c ✱ ♣♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❛❞♠✐t✐r s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ q✉❡ c ✱ xn = ts = u ✐♠♣❧✐❝❛♥❞♦ ❡♠ c ✱ ❧♦❣♦✱ t❡♠♦s ✭✐✐✮✳

  = ❈❛s♦ ✸✿ Z(G)\N = ∅ ❡ G {1, s}✳ x , ◆♦t❡ q✉❡ ♣❡❧❛ ❛✜r♠❛çã♦✱ c

  ∈ G ∀x ∈ G✳ ❙✉♣♦♥❤❛ ♣♦r ❛❜s✉r❞♦ q✉❡ ❡①✐st❡ y y = s ❡ s❡❥❛ x /∈ N❀ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ x /∈ G ∗ ✳ ❙❛❜❡♠♦s

  ∈ Z(G) ⊂ N t❛❧ q✉❡ c xy = c x c y (x, y) = s = 1 ∗ 2 q✉❡ c ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ xy ∈ G ⊂ Z(G)✱ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ∗ = = Z(G)\N = ∅✳ P♦rt❛♥t♦✱ ♥❡ss❡ ❝❛s♦✱ G Z(G) ❡ x {x, sx}✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ G é ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ∗✱ ♦✉ s❡❥❛✱ ✭✐✮ ♦❝♦rr❡✳

  = ❈❛s♦ ✹✿ Z(G)\N = ∅ ❡ G {1, s, t, u}✳

  = x / ❉❡s❞❡ q✉❡ G {1, s, t, u}✱ ❡♥tã♦ ❡①✐st❡ x / ∈ N t❛❧ q✉❡ c ∈ {1, s}✱ ♣♦✐s✱ ❝❛s♦ wy = c w c y (w, y) wy c w c y

  ❝♦♥trár✐♦✱ ❛ ❡q✉❛çã♦ c ✐♠♣❧✐❝❛r✐❛ ❡♠ (w, y) = c ∈ {1, s} , ∀w, y ∈

  x = t y = u y ✻✸ ❣❡♥❡r❛❧✐❞❛❞❡ q✉❡ c ✳ ❙❡ ❡①✐st✐r y ∈ G t❛❧ q✉❡ c ✱ ❡♥tã♦ ✭✐✐✮ ♦❝♦rr❡❀ s❡ c 6= u, ∀y ∈ G

  ✱ ❡♥tã♦ Z(G) ⊂ G ✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ ❝♦♠♦ Z(G) ⊂ N✱ ❡♥tã♦ ❡①✐st❡ n ∈ Z(G) ∩ N n = s nx = c n c x (n, x) = st = u t❛❧ q✉❡ c ✱ ❛ss✐♠✱ c ✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ❧♦❣♦✱ ❝♦♠♦ G ∗ ⊂ Z(G)✱

  = ❡♥tã♦ G ∗

  Z(G) ⊂ N ❡ ✭✐✐✐✮ ♦❝♦rr❡✳ x , c y ❋✐♥❛❧♠❡♥t❡✱ ♣❛r❛ ✈❡r✐✜❝❛r q✉❡ (x, y) = 1, ∀x, y t❛✐s q✉❡ c 6= s✱ s✉♣♦♥❤❛ x , c y

  ♣♦r ❛❜s✉r❞♦ q✉❡ (x, y) 6= 1✱ s❡♥❞♦ c 6= s✳ ◆❡st❡ ❝❛s♦✱ ❝♦♠♦ x, y / ∈ Z(G)✱ ❡♥tã♦ c x y xy = s 6= 1 6= c ✱ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ x, y /∈ N✱ ✐♠♣❧✐❝❛♥❞♦ ❡♠ c ✱ ❥á q✉❡ xy ∈ N\Z(G)✳ xy x , c y , (x, y)

  P❡❧❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✱ s = c ∈ {c } ❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ▲❡♠❛ ✹✳✼✱ ❛✜r♠❛ x y = (x, y) q✉❡ c ♦✉ c ✱ ❝♦♥tr❛r✐❛♥❞♦ ❛ ❤✐♣ót❡s❡✳ ❚❡♦r❡♠❛ ✹✳✶✹✳ ❙❡❥❛ R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ t❛❧ q✉❡ charR 6= 2✱ G ✉♠ ❣r✉♣♦ ♥ã♦ ❛❜❡❧✐❛♥♦ 2 ,

  ❝♦♠ ✐♥✈♦❧✉çã♦ ∗ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ✉♠❛ ♦r✐❡♥t❛çã♦ σ : G → {±1} ❡ t❛❧ q✉❡ x ∈ G ∀x ∈ G 2

  ✳ ❙❡ R 6= {0} ♦✉ (G ∩ Z(G))\N 6= ∅✱ ❡♥tã♦ RG é ❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ G

  ⊂ Z(G) ❡ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s ♦❝♦rr❡♠✿ 2 = c x x, x = 1 ∗ x

  ✭✐✮ x ∀x ∈ G✱ ♦♥❞❡ c ∈ G ∩ Z(G) ❡ c ✳ ❆❧é♠ ❞✐ss♦✱ G é ✉♠ 2✲❣r✉♣♦ xy = c x c y (x, y) xy x , c y , (x, y) ❛❜❡❧✐❛♥♦ ❡❧❡♠❡♥t❛r✱ c ❡ s❡ (x, y) 6= 1✱ ❡♥tã♦ c ∈ {c }✳ n =

  ✭✐✐✮ N é ❛❜❡❧✐❛♥♦✱ ♦✉ charR = 4 ❡ c ∈ N {1, s} , ∀n ∈ N✳ x n , (n, x)

  ✭✐✐✐✮ ❙❡ n ∈ N ❡ x /∈ N sã♦ t❛✐s q✉❡ (n, x) 6= 1✱ ❡♥tã♦ c ∈ {c }✳ x , c y ✭✐✈✮ ❙❡ x, y /∈ N sã♦ t❛✐s q✉❡ (x, y) 6= 1✱ ❡♥tã♦ (x, y) ∈ {c }✳ x y , c

  ❆❧é♠ ❞✐ss♦✱ s❡ N é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ❡♥tã♦ (x, y) = 1, ∀x, y t❛✐s q✉❡ c 6= s ❡ ♦❝♦rr❡ ✉♠ ❞♦s s❡❣✉✐♥t❡s✿

  ✭❛✮ G é ✉♠ ❙▲❈✲❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ∗✳ = 2 2 x = t y = u

  ✭❜✮ G {1, s, t, u} ≃ C × C ✱ ❡ ❡①✐st❡♠ x, y ∈ G t❛✐s q✉❡ c ❡ c ✳ 2 2 = x

  ✭❝✮ G {1, s, t, u} ≃ C × C ✱ N é ✉♠ ❙▲❈✲❣r✉♣♦✱ G ⊂ N✱ c ∈ {1, s, t} , ∀x ∈ G ❡ y = t ❡①✐st❡ y ∈ G t❛❧ q✉❡ c ✳

  ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ RG s❡❥❛ ❝♦♠✉t❛t✐✈♦✳ ❆ Pr♦♣r♦s✐çã♦ ✹✳✶✷ ❣❛r❛♥t❡ q✉❡ ❛ ✐♥✈♦❧✉çã♦ ♣♦ss✉✐ ❛ ❢♦r♠❛ ❞❡s❝r✐t❛ ❡♠ ✭✐✮✳ ❆❧é♠ ❞✐ss♦✱ ♦ ✐t❡♠ ✭✐✐✮ s❡❣✉❡ ❞♦ ❈♦r♦❧ár✐♦ ✹✳✻❀ ♦s ✐t❡♥s ✭✐✐✐✮ ❡ ✭✐✈✮ s❡❣✉❡♠ ❞♦s ▲❡♠❛s ✹✳✺ ❡ ✹✳✼✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ x )

  P❛r❛ ✈❡r✐✜❝❛r q✉❡ ❛ r❡❝í♣r♦❝❛✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ (x−σ(x)xc ❝♦♠✉t❛ ❝♦♠ (y y ) ∗ ∗

  − σ(y)yc ✱ ❝♦♠ x, y /∈ G ✱ ❥á q✉❡ G ⊂ Z(G)✳ ◆♦t❡ q✉❡ s❡ (x, y) = 1✱ ❛ ✈❡r✐✜❝❛çã♦ é ✐♠❡❞✐❛t❛✱ ❛ss✐♠✱ ♣♦❞❡♠♦s ❛❞♠✐t✐r q✉❡ (x, y) 6= 1✳

  ❙✉♣♦♥❤❛ q✉❡ x, y ∈ N✱ ♥❡ss❡ ❝❛s♦✱ N é ♥ã♦ ❛❜❡❧✐❛♥♦✱ ♣♦rt❛♥t♦ ♦ ✐t❡♠ ✭✐✐✮ ❡ ♦ ✐t❡♠ ✭✸✮ ❞♦ ❚❡♦r❡♠❛ ✹✳✶ ❣❛r❛♥t❡♠ ♦ r❡s✉❧t❛❞♦✳

  x = c y x = (x, y) ✻✹ ❙❡❥❛♠ x /∈ N ❡ y ∈ N✳ P❡❧♦ ✐t❡♠ ✭✐✐✐✮✱ c ♦✉ c ✳ ❙✉♣♦♥❤❛ q✉❡ c x = c y

  ✱ ❛ss✐♠ (x + xc x )(y x ) = xy + xyc x x

  − yc − xyc − xy = 0 = yx + yxc x x

  − yxc − yx = (y x )(x + xc x ). x = (x, y) y = c xy − yc ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ c ✱ ❧♦❣♦ c ✱ ❛ss✐♠

  (x + xc x )(y y ) = xy + xyc x y x c y − yc − xyc − xyc

  = yx(x, y) + yxc x (x, y) y (x, y) x c y (x, y) − yxc − yxc

  = yxc x + yx y c x y − yxc − yxc = (y y )(x + xc x ).

  − yc x , c y x ❙❡❥❛♠ ❛❣♦r❛ x, y /∈ N✳ P❡❧♦ ✐t❡♠ ✭✐✐✐✮✱ (x, y) ∈ {c }✳ ❙✉♣♦♥❤❛ q✉❡ (x, y) = c ✱

  ❛ss✐♠✱ (x + xc x )(y + yc y ) = xy + xyc x + xyc y + xyc x c y

  = yx(x, y) + yxc x (x, y) + yxc y (x, y) + yxc x c y (x, y) = yxc x + yx + yxc y c x + yxc y y = (y + yc y )(x + xc x ).

  ❙❡ (x, y) = c ✱ ♣♦❞❡♠♦s ♣r♦❝❡❞❡r ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✳ ❉❡ss❛ ❢♦r♠❛ ❡♥❝♦♥tr❛♠♦s q✉❡ RG é ❝♦♠✉t❛t✐✈♦✳

  ❆s ♣r♦♣r✐❡❞❛❞❡s r❡s✉❧t❛♥t❡s ❞♦ ❝❛s♦ N ♥ã♦ ❛❜❡❧✐❛♥♦ s❡❣✉❡♠ ❞❛ Pr♦♣♦s✐çã♦ ✹✳✶✸✳ 2 2 = ❊①❡♠♣❧♦ ✹✳✶✺✳ ❙❡❥❛ N = H ❞❛❞♦ ♥♦ ❊①❡♠♣❧♦ ✶✳✶✹ ❡ ❞❡✜♥❛ J = N ⋊ C ✱ ❝♦♠ C hyi 1 2 2 2

  ) = (y, x ) = x x = t 1 2 ❡ (y, x ✳ c

  ❆ ♣❛rt✐r ❞❛ ♣r♦♣r✐❡❞❛❞❡ (a, bc) = (a, c)(a, b) ✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✱ ♣❛r❛ i, j ∈ {1, 2}✱ x x j j 2 2 (y, x i x j ) = (y, x j )(y, x i ) = tt = t = 1, 2 2

  = 1 ❥á q✉❡ t ∈ Z(J) ❡ t ✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ N é ✉♠ ❙▲❈✲❣r✉♣♦✱ N/Z(N) ≃ C 1 2 1 2

  × C , x , x x

  ❡ τ = {1, x } é ✉♠ tr❛♥s✈❡rs❛❧ ❞❡ Z(N) ❡♠ N✱ ♦✉ s❡❥❛✱ s❡ n ∈ Z(N)✱ ❡♥tã♦ 1 2 n , x é ✉♠❛ ♣❛❧❛✈r❛ ♥♦ ❛❧❢❛❜❡t♦ {x } ❝♦♠ ✉♠❛ q✉❛♥t✐❞❛❞❡ ♣❛r ❞❡ ❧❡tr❛s✳ ❉❡ss❛ ❢♦r♠❛✱

  ( Z(N), y) = {1} ❡✱ ❛ss✐♠✱ Z(N) = Z(J)✱ ❥á q✉❡ (J : N) = 2✳

  ❈♦♠♦ N é ✉♠ ❙▲❈✲❣r✉♣♦✱ ♣♦❞❡♠♦s ♠✉♥✐✲❧♦ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ∗ ❡ ❡s✲ = yt t❡♥❞❡r ❡ss❛ ✐♥✈♦❧✉çã♦ ❛ J ❛ ♣❛rt✐r ❞❡ y ✳ ❆❧é♠ ❞✐ss♦✱ ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛♠♦s q✉❡ ❛

  ❛♣❧✐❝❛çã♦ σ✱ s❡♥❞♦ σ(n) = 1✱ s❡ n ∈ N ❡ σ(y) = −1✱ ♣♦❞❡ s❡r ❡st❡♥❞✐❞❛ ❛ ✉♠❛ ♦r✐❡♥t❛çã♦ ❞❡ J ❝✉❥♦ ♥ú❝❧❡♦ é ♣r❡❝✐s❛♠❡♥t❡ N ❡ ❡st❛ é ❝♦♠♣❛tí✈❡❧ ❝♦♠ ∗✳

  ✻✺ ✉♠ ❙▲❈✲❣r✉♣♦ ✈❡r✐✜❝❛ ♦ ✐t❡♠ ✭✐✐✮✳ a n , (a, n)

  ❱❛♠♦s ♠♦str❛r q✉❡ ♦ ✐t❡♠ ✭❝✮ é s❛t✐s❢❡✐t♦✱ ♦✉ s❡❥❛ c ∈ {c }✱ ♣❛r❛ a / ∈ N

  ✱ n ∈ N ❝♦♠ (n, a) 6= 1✳ ❈♦♠♦ (J : N) = 2✱ t❡♠♦s q✉❡ a = ym✱ ❝♦♠ m ∈ N❀ xy = c x c y (x, y) ❛❞❡♠❛✐s✱ ❝♦♠♦ ❡❧❡♠❡♥t♦s ❝❡♥tr❛✐s sã♦ s✐♠étr✐❝♦s✱ ❛ ♣r♦♣r✐❡❞❛❞❡ c ❣❛r❛♥t❡ xz = c x , q✉❡ c ❡ (xz, t) = (x, t) ∀x, t ∈ J ❡ z ∈ Z(J)✱ ♣♦rt❛♥t♦ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r q✉❡ 1 2 1 2 m, n , x , x x

  ∈ τ = {1, x } ❡ n 6= 1✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦ (a, n) = 1✱ ❝♦♥tr❛r✐❛♥❞♦ ❛ n = s ❤✐♣ót❡s❡❀ ❧♦❣♦ c ✳ ❙❛❜❡♠♦s q✉❡ c a = c ym = c y c m (y, m) = tc m (y, m)

  ❡ (a, n) = (ym, n) = (y, n)(m, n),

  ❆♥❛❧✐s❛♥❞♦ t♦❞❛s ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ m ❡ n r❡st❛♥t❡s ❡ q✉❡ ♥ã♦ ✐♠♣❧✐q✉❡♠ ❡♠ (a, n) = 1 a n , (a, n) ✱ s❡♠♣r❡ ❡♥❝♦♥tr❛r❡♠♦s q✉❡ c ∈ {s, (a, n)} = {c }✳

  P❛r❛ ✈❡r✐✜❝❛r ✭❞✮✱ s✉♣♦♥❤❛ q✉❡ a, b /∈ N ♥ã♦ ❝♦♠✉t❛♠✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛ ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ a = ym ❡ b = yn ❝♦♠ n, m ∈ τ✳ ◆♦t❡ q✉❡ 1 2

  , x {n, m} ∩ {x } 6= ∅✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦ (a, b) = 1✳ ❙❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ♣♦❞❡♠♦s 1

  ❛ss✉♠✐r q✉❡ n = x ✳ ◆♦t❡ q✉❡ (a, b) = (ym, yx 1 ) = (x 1 , m)(y, m)(y, x 1 ) = (x 1 , m)(y, m)t, c a = c yx = c y c x (y, x 1 1 ❡ 1 ) = tst = s, c b = c ym = c y c m (y, m) = tc m (y, m).

  ❆♥❛❧✐s❛♥❞♦ ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s r❡st❛♥t❡s✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♦ ✐t❡♠ ✭❞✮ s❡ ✈❡r✐✜❝❛✳ ❉❡ss❛ ❢♦r♠❛ t♦♠❛♥❞♦ R ✉♠ ❛♥❡❧ t❛❧ q✉❡ charR = 4✱ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✹✳✶✹✱ σ t❡♠♦s q✉❡ RJ ∗ é ❝♦♠✉t❛t✐✈♦✳ ◆❡st❡ ❝❛s♦✱ ✈❛❧❡ ♦❜s❡r✈❛r q✉❡ J ❡ ∗ ✈❡r✐✜❝❛♠ ❛ ❝♦♥❞✐çã♦ ✭❝✮ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳

  ❈❛♣ít✉❧♦ ✺ ❈♦♥❝❧✉sõ❡s ✺✳✶ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s

  ❖ ♣r♦♣ós✐t♦ ❞❛ ♣r❡s❡♥t❡ t❡s❡ ❝♦♥s✐st❡ ❡♠ ❡st❡♥❞❡r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❛❝❡r❝❛ ❞❛ r❡❧❛✲ çã♦ ❡♥tr❡ ❛s ❡str✉t✉r❛s ❜❛s❡ ❞❡ ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG ❡ ❝❡rt❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s✱ ❛ s❛❜❡r ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡✱ ✈❡r✐✜❝❛❞❛s ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s s✐♠é✲ tr✐❝♦s ❡ ❛♥t✐ss✐♠étr✐❝♦s ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛ ♥❡st❡ ❛♥❡❧ ❞❡ ❣r✉♣♦✳

  ◆❡st❡ s❡♥t✐❞♦✱ ❛❧❝❛♥ç❛♠♦s ♥♦ ❈❛♣ít✉❧♦ ✷ ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ♦ ❝♦♥❥✉♥t♦ RG s❡❥❛ ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ❚❡♦r❡♠❛ ✷✳✶✹✳ ❆❧é♠ ❞✐ss♦✱ ✈❡r✐✜❝❛♠♦s q✉❡ ♣❛r❛ t❛❧ ❝❛r❛❝t❡r✐③❛çã♦ ❢♦✐ ♥❡❝❡ssár✐♦ ❛♣❡♥❛s ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣ré✈✐♦ ❞♦ ❛♥á❧♦❣♦ ♣❛r❛ ❛s ✐♥✈♦❧✉çõ❡s ♥ã♦ ♦r✐❡♥t❛❞❛s✱ ❞♦♥❞❡ ♣♦❞❡rí❛♠♦s t❡r ❝❧❛ss✐✜❝❛❞♦ ♦ ❝❛s♦ ❝♦♠ ❛ ♦r✐❡♥t❛çã♦ ❝❧áss✐❝❛✱ ❈♦r♦❧ár✐♦ ✷✳✶✻✱ ❝♦♠♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✹✳ ◆♦ ♠❡s♠♦ ❝❛♣ít✉❧♦✱ ❝♦♠ ❛❧❣✉♠❛s ❝♦♥❞✐çõ❡s ❛❞✐❝✐♦♥❛✐s✱ ♣✉❞❡♠♦s ❡①tr❛✐r ✐♥❢♦r♠❛çõ❡s ✐♠♣♦rt❛♥t❡s ❛❝❡r❝❛ ❞❛ ❝❛r❛❝t❡ríst✐❝❛ ❞♦ ❛♥❡❧ R ❡ ❞♦ ❡①♣♦❡♥t❡ ❞♦ ❣r✉♣♦ q✉♦❝✐❡♥t❡ G/N✱ ❈♦r♦❧ár✐♦ ✷✳✶✺✱ ♦✉ ❛té ❞❡s❝r❡✈❡r ❝♦♠♣❧❡t❛♠❡♥t❡ ❛ ♦r✐❡♥t❛çã♦ σ✱ Pr♦♣♦s✐çã♦ ✷✳✶✼✳

  ◆♦ ❈❛♣ít✉❧♦ ✸ ❛❧❝❛♥ç❛♠♦s ✉♠ r❡s✉❧t❛❞♦ s❡♠❡❧❤❛♥t❡ ❛♦ ❞♦ ❈❛♣ít✉❧♦ ✷ ♣❛r❛ q✉❡ ♦s s✐♠étr✐❝♦s s❡❥❛♠ ❛♥t✐❝♦♠✉t❛t✐✈♦s✱ ❚❡♦r❡♠❛ ✸✳✶✼✳ ❊♥tr❡t❛♥t♦✱ ♥❛ ♣r❡s❡♥t❡ s✐t✉❛çã♦✱ ❝❧❛r❛♠❡♥t❡ ♣❡r❝❡❜❡✲s❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ❛♥á❧♦❣♦ ♣❛r❛ ❛s ✐♥✈♦❧✉çõ❡s ♦r✐✲ ❡♥t❛❞❛s ✉s✉❛✐s✳ ❊♥❝♦♥tr❛♠♦s t❛♠❜é♠ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❞❡st❛q✉❡ ❛❝❡r❝❛ ❞❡ R ❡ G/N ❝❛s♦ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s ❛❞✐❝✐♦♥❛✐s s❡❥❛♠ ❛❞♠✐t✐❞❛s✱ ✈✐❞❡ ❈♦r♦❧ár✐♦ ✸✳✶✽✳ ❯♠ ❢❛t♦ ❞❡ ❞❡st❛q✉❡ ♥♦ t❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ❞❡st❡ ❝❛♣ít✉❧♦ é ♦ ❢❛t♦ ❞❡ q✉❡ ❛ ❝❧❛ss❡ ❞❡ ❣r✉♣♦s s♦❧✉çõ❡s ♣❛r❛ ♦ ❝❛s♦ ❛♥á❧♦❣♦ ❝♦♠ ✐♥✈♦❧✉çõ❡s ♥ã♦ ♦r✐❡♥t❛❞❛s é ♦ ♠❡s♠♦ ♣❛r❛ ❛s ✐♥✈♦❧✉çõ❡s ❝♦♠ ♦r✐❡♥t❛çã♦ ✉s✉❛❧✱ ❡♥tr❡t❛♥t♦✱ ♥♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧ tr❛t❛❞♦ ♥❛ t❡s❡✱ ✉♠❛ ♥♦✈❛ ❝❧❛ss❡ s✉r❣❡ ❞❡♥tr♦ ❞❡st❛s s♦❧✉çõ❡s✳

  ❉❡✈✐❞♦ ❛ ❛✉sê♥❝✐❛ ♥❛ ❧✐t❡r❛t✉r❛ ❞❡ ✉♠ ❡st✉❞♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❛♥t✐ss✐♠é✲ tr✐❝♦s s♦❜ ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s ✉s✉❛✐s ❡ ❛ ❞✐✜❝✉❧❞❛❞❡ ❡♠ tr❛t❛r ❡ss❡ t❡♠❛✱ ❛♥t❡s ❞❡ ♣❛ss❛r ♣❛r❛ ❛s ♦r✐❡♥t❛çõ❡s ❣❡♥❡r❛❧✐③❛❞❛s✱ é ♣r❛t✐❝❛♠❡♥t❡♥t❡ ✐♠♣r❡s❝✐♥❞í✈❡❧ ❝❛r❛❝t❡r✐③❛r ❡ss❡ ❝❛s♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♥♦ ❈❛♣ít✉❧♦ ✹ ✐♥✐❝✐❛♠♦s ❡ss❡ ❡st✉❞♦ ♣❛r❛ ✉♠❛ ♦r✐❡♥t❛çã♦ ✉s✉❛❧✱ ❛❧❝❛♥ç❛♥❞♦

  ✻✼ ✉♠ r❡s✉❧t❛❞♦ ♣❛r❝✐❛❧ ❛♦ ❛❞♠✐t✐r q✉❡ R ♣♦ss✉✐ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ t♦rsã♦ ✷ ♦✉ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ❡♠ G\N q✉❡ s❡❥❛ s✐♠étr✐❝♦ ❡ ❝❡♥tr❛❧✳

  ❆♣ós ♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ❛❝✐♠❛✱ ♦ ❡st❛❞♦ ❞❛ ❛rt❡ ❞♦ ❡st✉❞♦ ❞❛ ❝♦♠✉t❛✲ t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s s✐♠étr✐❝♦s ❡ ❛♥t✐ss✐♠étr✐❝♦s s♦❜ ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s ♣♦❞❡ s❡r r❡s✉♠✐❞♦ ♥❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿

  ❈♦♠✉t❛t✐✈✐❞❛❞❡ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ Pr♦♣r✐❡❞❛❞❡s ❞❡ ▲✐❡ RG σ

  ✲ ❈❛♣ít✉❧♦ ✸ ❬❱✶✸❪

  RG σ ∗ ✲ ❈❛♣ít✉❧♦ ✷ ❚❛❜❡❧❛ ✺✳✶✿ ■❞❡♥t✐❞❛❞❡s ♣❛r❛ ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s

  ❆❧é♠ ❞✐ss♦✱ ❛ ❚❛❜❡❧❛ ✶✳✸✱ q✉❡ ✐❧✉str❛ ❛s ✐❞❡♥t✐❞❛❞❡s ♣❛r❛ ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s ✉s✉❛✐s✱ ♣❛ss❛ ❛ t❡r ❛ ❝♦♥✜❣✉r❛çã♦ ❛ s❡❣✉✐r✿

  ❈♦♠✉t❛t✐✈✐❞❛❞❡ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ Pr♦♣r✐❡❞❛❞❡s ❞❡ ▲✐❡ RG σ ∗

  ❬❇P✵✻✱ ●P✶✸❜❪ ❬●P✶✹❪ ❬❈P✶✷✱ ❱✶✸❪

  RG σ ∗ ❬❇❏❘✵✼❪✱ ❈❛♣ít✉❧♦ ✹ ❬●P✶✸❜❪ ❚❛❜❡❧❛ ✺✳✷✿ ■❞❡♥t✐❞❛❞❡s ♣❛r❛ ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s

  ✺✳✷ ❚r❛❜❛❧❤♦s ❋✉t✉r♦s

  ❉❡✈✐❞♦ ❛♦s r❡s✉❧t❛❞♦s ❝✐t❛❞♦s ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♥♦ ❝❛♠✐♥❤♦ ❛❜♦r❞❛❞♦ ♣♦r ❡st❡s tr❛❜❛❧❤♦s✱ ♣♦✉❝♦ t❡♠✲s❡ ❛ ❛❝r❡s❝❡♥t❛r ♥♦ ❡st✉❞♦ ❞❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❡❧❡♠❡♥t♦s s✐♠étr✐❝♦s ❡ ❛♥t✐ss✐♠étr✐❝♦s✱ ✈✐st♦ q✉❡ ♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧ ❞❡ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❢♦✐ ❛❧❝❛♥ç❛❞♦✳

  ❖❜s❡r✈❛♥❞♦ ❛s ❧❛❝✉♥❛s ❞❛ ❚❛❜❡❧❛ ✺✳✶✱ ✐♠❡❞✐❛t❛♠❡♥t❡ ❝♦♥❝❧✉í♠♦s q✉❡ ✉♠ ❞♦s σ ♣♦ssí✈❡✐s t❡♠❛s ❛ s❡r tr❛t❛❞♦ ❢✉t✉r❛♠❡♥t❡ ❝♦♥s✐st❡ ❡♠ ❡st✉❞❛r ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❡ RG σ ❡ RG ∗ ✳ ❆❧é♠ ❞✐ss♦✱ ❛❧❣✉♥s ❝❛s♦s ❥á ❡st✉❞❛❞♦s ❡ ❡①✐❜✐❞♦s ♥❛s ❞✉❛s t❛❜❡❧❛s ❛❝✐♠❛✱ ♠❡r❡❝❡♠ ❛✐♥❞❛ ❛t❡♥çã♦✱ ✈✐st♦ ❛s s❡❣✉✐♥t❡s ♦❜s❡r✈❛çõ❡s✿

  ✶✳ ❖ ❡st✉❞♦ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ▲✐❡ r❡❛❧✐③❛❞❛s ❡♠ ❬❱✶✸❪✱ ❛♣♦♥t❛❞❛s ♣❡❧❛ ❚❛❜❡❧❛ ✺✳✶✱ ❢♦✐ ❜❛s❡❛❞♦ ♥❛ ❜✉s❝❛ ❞❡ ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❛ ✐❞❡♥t✐❞❛❞❡ ♣♦ss❛ s❡r ❣❛r❛♥t✐❞♦✱ ❞❡✐①❛♥❞♦ ❡♠ ❛❜❡rt♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ s❛❜❡r q✉❛✐s ❛s ❝♦♥❞✐çõ❡s s♦❜r❡ R ❡ G σ ∗ ♣❛r❛ q✉❡ RG ❡ RG σ s❛t✐s❢❛ç❛♠ t❛✐s ♣r♦♣r✐❡❞❛❞❡s✳ ❆❧é♠ ❞♦ ♠❛✐s✱ ♥❡st❡ tr❛❜❛❧❤♦ ❛ ✐♥✈♦❧✉çã♦ ∗ ❢♦✐ t♦♠❛❞❛ ❝♦♠♦ ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✱ ❞♦♥❞❡ ❝♦♥❝❧✉í♠♦s q✉❡ ❛ ❜✉s❝❛ ❞❡ ❝♦♥❞✐çõ❡s ♣❛r❛ ♦ ❧❡✈❛♥t❛♠❡♥t♦ s❡r ❣❛r❛♥t✐❞♦✱ ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛ q✉❛❧q✉❡r✱ ♠❡r❡❝❡ ❛t❡♥çã♦✳

  ✷✳ ❖s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s ❡♠ ❬❇❏❘✵✼❪ ❝❛r❛❝t❡r✐③❛♠ ❝♦♠♣❧❡t❛♠❡♥t❡ ♦ s❡✉ t❡♠❛ ♣r♦✲

  ✻✽ ❝❧áss✐❝❛✳ ❈♦♠♣❧❡♠❡♥t❛♥❞♦ s❡✉ ❡st✉❞♦✱ ♥♦ ❈❛♣ít✉❧♦ ✹✱ t❡♠♦s ❛ ❛❜♦r❞❛❣❡♠ ❞❡st❡ ❝❛s♦ ♣❛r❛ ∗ s❡♥❞♦ ✉♠❛ ✐♥✈♦❧✉çã♦ q✉❛❧q✉❡r✱ ♣♦ré♠ ❡st❡s ❞♦✐s r❡s✉❧t❛❞♦s ❛✐♥❞❛ ❞❡✐①❛♠ ❡♠ ❛❜❡rt♦ ❛❧❣✉♠❛s q✉❡stõ❡s✱ ✈✐st♦ q✉❡ ♥❡st❡ ú❧t✐♠♦ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s ❛❞✐❝✐♦♥❛✐s sã♦ ❛ss✉♠✐❞❛s✳

  ✸✳ ◆♦✈❛♠❡♥t❡ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❡♠ ❬❈P✶✷❪ ♦s ❛✉t♦r❡s ❝♦♠♣❧❡t❛♠ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞♦ s❡✉ r❡s♣❡❝t✐✈♦ ❝❛s♦ q✉❛♥❞♦ ∗ é ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✳ ❊♠ ❬❱✶✸❪✱ ❛❧❣✉♠❛s r❡s♣♦st❛s ♣❛r❛ ♦ ❝❛s♦ ♠❛✐s ❣❡r❛❧ sã♦ ❞❛❞❛s✱ ♣♦ré♠ ❡st❛s ♥ã♦ ❡①t✐♥❣✉❡♠ t♦❞❛s ❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ❞♦ ♣r♦❜❧❡♠❛ ❡♠ q✉❡stã♦✳

  ❘❡❢❡rê♥❝✐❛s

  ❬❆✻✾❪ ❆▼■❚❙❯❘✱ ❙✳ ❆✳ ■❞❡♥t✐t✐❡s ✐♥ r✐♥❣s ✇✐t❤ ✐♥✈♦❧✉t✐♦♥s✳ ■sr❛❡❧ ❏✳ ▼❛t❤✳ ✈✳ ✼✱ ♣✳✻✸✲✻✽✱ ✶✾✻✾✳

  ❬❇❏P❘✵✾❪ ❇❘❖❈❍❊ ❈❘■❙❚❖✱ ❖✳❀ ❏❊❙P❊❘❙✱ ❊✳❀ P❖▲❈■◆❖ ▼■▲■❊❙✱ ❈✳❀ ❘❯■❩✱ ▼✳ ❆♥t✐s②♠♠❡tr✐❝ ❡❧❡♠❡♥ts ✐♥ ❣r♦✉♣ r✐♥❣s ■■✱ ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✈✳ ✽✱ ♥✳ ✶✱ ♣✳ ✶✶✺✲✶✷✼✱ ✷✵✵✾✳

  ❬❇❏❘✵✼❪ ❇❘❖❈❍❊ ❈❘■❙❚❖✱ ❖✳❀ ❏❊❙P❊❘❙✱ ❊✳❀ ❘❯■❩ ▼❆❘❮◆✱ ▼✳ ❆♥t✐s②♠♠❡tr✐❝ ❡❧❡✲ ♠❡♥ts ✐♥ ❣r♦✉♣ r✐♥❣s ✇✐t❤ ❛♥ ♦r✐❡♥t❛t✐♦♥ ♠♦r♣❤✐s♠✳ ❛r❳✐✈✿✵✼✵✺✳✸✶✵✻ ❬♠❛t❤✳❑❚❪✳

  ❬❇P✵✻❪ ❇❘❖❈❍❊ ❈❘■❙❚❖✱ ❖✳❀ P❖▲❈■◆❖ ▼■▲■❊❙✱ ❈✳ ❙②♠♠❡tr✐❝ ❡❧❡♠❡♥ts ✉♥❞❡r ♦r✐❡♥✲ t❛t❡❞ ✐♥✈♦❧✉t✐♦♥s ✐♥ ❣r♦✉♣ r✐♥❣s✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✈✳ ✸✹✱ ♥✳ ✾✱ ♣✳ ✸✸✹✼✲✸✸✺✻✱ ✷✵✵✻✳

  ❬❈P✶✷❪ ❈❆❙❚■▲▲❖ ●Ó▼❊❩✱ ❏✳ ❍✳❀ P❖▲❈■◆❖ ▼■▲■❊❙✱ ❈✳ ▲✐❡ Pr♦♣❡rt✐❡s ♦❢ ❙②♠♠❡tr✐❝ ❊❧❡♠❡♥ts ❯♥❞❡r ❖r✐❡♥t❡❞ ■♥✈♦❧✉t✐♦♥s✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✈✳ ✹✵✱ ♥✳ ✶✷✱ ♣✳ ✹✹✵✹✲✹✹✶✾✱ ✷✵✶✷✳

  ❬●❏P✾✻❪ ●❖❖❉❆■❘❊✱ ❊✳ ●✳❀ ❏❊❙P❊❘❙✱ ❊✳❀ P❖▲❈■◆❖ ▼■▲■❊❙✱ ❈✳ ❆❧t❡r♥❛t✐✈❡ ▲♦♦♣ ❘✐♥❣s✱ ❊❧s❡✈✐❡r✱ ✶✾✾✻✳

  ❬●P✶✸❛❪ ●❖❖❉❆■❘❊✱ ❊✳ ●✳❀ P❖▲❈■◆❖ ▼■▲■❊❙✱ ❈✳ ■♥✈♦❧✉t✐♦♥s ❛♥❞ ❆♥t✐❝♦♠♠✉t❛t✐✈✐t② ✐♥ ●r♦✉♣ ❘✐♥❣s✳ ❊♠✿ ❈❛♥❛❞✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❇✉❧❧❡t✐♥✱ ✈✳ ✺✷✱ ♥✳ ✷✱ ♣✳ ✸✹✹✲✸✺✸✱ ✷✵✶✸✳

  ❬●P✶✸❜❪ ●❖❖❉❆■❘❊✱ ❊✳ ●✳❀ P❖▲❈■◆❖ ▼■▲■❊❙✱ ❈✳ ❖r✐❡♥t❡❞ ■♥✈♦❧✉t✐♦♥s ❛♥❞ ❙❦❡✇✲ s②♠♠❡tr✐❝ ❊❧❡♠❡♥ts ✐♥ ●r♦✉♣ ❘✐♥❣s✳ ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛ ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✈✳ ✶✷✱ ♥✳ ✶✱ ✷✵✶✸✳

  ❬●P✶✹❪ ●❖❖❉❆■❘❊✱ ❊✳ ●✳❀ P❖▲❈■◆❖ ▼■▲■❊❙✱ ❈✳ ❖r✐❡♥t❡❞ ●r♦✉♣ ■♥✈♦❧✉t✐♦♥s ❛♥❞ ❆♥t✐❝♦♠♠✉t❛t✐✈✐t② ✐♥ ●r♦✉♣ ❘✐♥❣s✳ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✈✳ ✹✷✱ ♥✳ ✹✱ ♣✳ ✶✻✺✼✲ ✶✻✻✼✱ ✷✵✶✹✳

  ❬●P❙✵✾❪ ●■❆▼❇❘❯◆❖✱ ❆✳❀ P❖▲❈■◆❖ ▼■▲■❊❙✱ ❈✳❀ ❙❊❍●❆▲✱ ❙✳ ❑✳ ▲✐❡ ♣r♦♣❡rt✐❡s ♦❢ s②♠♠❡tr✐❝ ❡❧❡♠❡♥ts ✐♥ ❣r♦✉♣ r✐♥❣s✱ ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛✱ ✈✳ ✸✷✶✱ ♥✳ ✸✱ ♣✳ ✽✾✵✲✾✵✷✱ ✷✵✵✾✳

  ✼✵ ❬●❙✽✾❪ ●■❆▼❇❘❯◆❖✱ ❆✳❀ ❙❊❍●❆▲✱ ❙✳ ❑✳ ❆ ▲✐❡ ♣r♦♣❡rt✐❡ ✐♥ ❣r♦✉♣ r✐♥❣s✱ Pr♦❝❡❡❞✐♥❣s

  ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✈✳ ✶✵✺✱ ♥✳ ✷✱ ♣✳ ✷✽✼✲✷✾✷✱ ✶✾✽✾ ❬●❙✾✸❪ ●■❆▼❇❘❯◆❖✱ ❆✳❀ ❙❊❍●❆▲✱ ❙✳ ❑✳ ▲✐❡ ◆✐❧♣♦t❡♥❝❡ ♦❢ ●r♦✉♣ ❘✐♥❣s✱ ❈♦♠♠✉♥✐❝❛✲ t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✈✳ ✷✶✱ ♥✳ ✶✶✱ ♣✳ ✹✷✺✸✲✹✷✻✶✱ ✶✾✾✸✳ ❬❍✼✻❪ ❍❊❘❙❚❊■◆✱ ■✳ ◆✳ ❘✐♥❣s ✇✐t❤ ✐♥✈♦❧✉t✐♦♥s✱ ❈❤✐❝❛❣♦✿ ❯♥✐✈❡rs✐t② ♦❢ ❈❤✐❝❛❣♦ Pr❡ss✱

  ✶✾✼✻✳ ✭❈❤✐❝❛❣♦ ❧❡❝t✉r❡s ✐♥ ♠❛t❤❡♠❛t✐❝s✮ ❬❏❘✵✺❪ ❏❊❙P❊❘❙✱ ❊✳❀ ❘✉✐③✱ ▼✳ ❆♥t✐s②♠♠❡tr✐❝ ❡❧❡♠❡♥ts ✐♥ ❣r♦✉♣ r✐♥❣s✱ ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛

  ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ✈✳ ✹✱ ♥✳ ✹✱ ♣✳ ✸✹✶✲✸✺✸✱ ✷✵✵✺✳ ❬❏❘✵✻❪ ❏❊❙P❊❘❙✱ ❊✳❀ ❘❯■❩ ▼❆❘❮◆✱ ▼✳ ❖♥ s②♠♠❡tr✐❝ ❡❧❡♠❡♥ts ❛♥❞ s②♠♠❡tr✐❝ ✉♥✐ts ✐♥

  ❣r♦✉♣ r✐♥❣s✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✈✳ ✸✹✱ ♥✳ ✷✱ ♣✳ ✼✷✼✲✼✸✻✱ ✷✵✵✻✳ ❬❑✾✽❪ ❑◆❯❙✱ ▼✲❆ ❡t ❛❧✳ ❚❤❡ ❇♦♦❦ ♦❢ ■♥✈♦❧✉t✐♦♥s✱ ❆▼❙ ❈♦❧❧♦q✉✐✉♠ P✉❜❧✐❝❛t✐♦♥s✱ ❱♦❧✳

  ✹✹✱ ✶✾✾✽✳ ❬▲✾✾❪ ▲❊❊✱ ●✳ ❚✳ ●r♦✉♣ ❘✐♥❣s ❲❤♦s❡ ❙②♠♠❡tr✐❝ ❊❧❡♠❡♥ts ❛r❡ ▲✐❡ ◆✐❧♣♦t❡♥t✱ Pr♦❝❡❡❞✐♥❣s

  ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✈✳ ✶✷✼✱ ♥✳ ✶✶✱ ♣✳ ✸✶✺✸✲✸✶✺✾✱ ✶✾✾✾✳ ❬▲✵✵❪ ▲❊❊✱ ●✳ ❚✳ ❚❤❡ ▲✐❡ n✲❊♥❣❡❧ Pr♦♣❡rt② ✐♥ ●r♦✉♣ ❘✐♥❣s✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱

  ✈✳ ✷✽✱ ♥✳ ✷✱ ♣✳ ✽✻✼✲✽✽✶✱ ✷✵✵✵✳ ❬▲✶✵❪ ▲❊❊✱ ●✳ ❚✳ ●r♦✉♣ ■❞❡♥t✐t✐❡s ♦♥ ❯♥✐ts ❛♥❞ ❙②♠♠❡tr✐❝ ❯♥✐ts ♦❢ ●r♦✉♣ ❘✐♥❣s✱ ◆❡✇

  ❨♦r❦✿ ❙♣r✐♥❣❡r✱ ✷✵✶✵✳ ❬▲❙❙✵✾❪ ▲❊❊✱ ●✳ ❚✳❀ ❙❊❍●❆▲✱ ❙✳ ❑✳❀ ❙P■◆❊▲▲■✱ ❊✳ ▲✐❡ ♣r♦♣❡rt✐❡s ♦❢ s②♠♠❡tr✐❝ ❡❧❡♠❡♥ts

  ✐♥ ❣r♦✉♣ r✐♥❣s ■■✱ ❏♦✉r♥❛❧ ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ❆❧❣❡❜r❛✱ ✈✳ ✷✶✸✱ ♥✳ ✻✱ ♣✳ ✶✶✼✸✲✶✶✼✽✱ ✷✵✵✾✳

  ❬◆✼✵❪ ◆❖❱■❑❖❱✱ ❙✳ P✳ ❆❧❣❡❜r❛✐❝ ❝♦♥str✉❝t✐♦♥ ❛♥❞ ♣r♦♣❡rt✐❡s ♦❢ ❤❡r♠✐t✐❛♥ ❛♥❛❧♦❣✉❡s ♦❢ ❑✲t❤❡♦r② ♦✈❡r r✐♥❣s ✇✐t❤ ✐♥✈♦❧✉t✐♦♥ ❢r♦♠ ✈✐❡✇♣♦✐♥t ♦❢ ❍❛♠✐❧t♦♥✐❛♥ ❢♦r♠❛❧✐s♠✱ ❆♣♣❧✐✲ ❝❛t✐♦♥s t♦ ❞✐✛❡r❡♥t✐❛❧ t♦♣♦❧♦❣② ❛♥❞ t❤❡ t❤❡♦r② ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ❝❧❛ss❡s ■■✱ ▼❛t❤❡♠❛t✐❝s ♦❢ t❤❡ ❯❙❙❘✲■③✈❡st✐②❛✱ ✈✳ ✹✱ ♥✳ ✷✱ ♣✳ ✷✺✼✲✷✾✷✱ ✶✾✼✵✳

  ❬PP❙✼✸❪ P❆❙❙■✱ ■✳ ❇✳ ❙✳❀ P❆❙❙▼❆◆✱ ❉✳ ❙✳❀ ❙❊❍●❆▲✱ ❙✳ ❑✳ ▲✐❡ s♦❧✈❛❜❧❡ ❣r♦✉♣ r✐♥❣s✱ ❈❛♥❛❞✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✈✳ ✷✺✱ ♣✳ ✼✹✽✲✼✺✼✱ ✶✾✼✸✳

  ❬❱✶✸❪ ❍❖▲●❯❮◆ ❱■▲▲❆✱ ❆✳ ■♥✈♦❧✉çõ❡s ❞❡ ❣r✉♣♦ ♦r✐❡♥t❛❞❛s ❡♠ á❧❣❡❜r❛s ❞❡ ❣r✉♣♦✳ ✷✵✶✸✳ ✼✻❢✳ ❚❡s❡ ✭❉♦✉t♦r❛❞♦✮ ✲ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛✱ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✱ ❙ã♦ P❛✉❧♦✱ ✷✵✶✸✳

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