Darstellungstheorie endlicher Gruppen by Peter Müller

By Peter Müller

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N s✐♥❞ ❞✐❡ n ❊❧❡♠❡♥t❡ τ1 τj ♣❛❛r✇❡✐s❡ ✈❡rs❝❤✐❡❞❡♥✳ ❆❜❡r |F | = n✱ ❞❛❤❡r ❜❡st❡❤t F ❣❡♥❛✉ ❛✉s ❞✐❡s❡♥ ❊❧❡♠❡♥t❡♥✳ ❆❜❡r ❞❛♠✐t s❡❤❡♥ ✇✐r✱ ❞❛ss F ♠✉❧t✐♣❧✐❦❛t✐✈ −1 ❛❜❣❡s❝❤❧♦ss❡♥ ✐st✿ ❙❡✐❡♥ a, b ∈ F ✳ ❉❛♥♥ ❣✐❧t ❛✉❝❤ a ∈ F ✱ ✉♥❞ ♥❛❝❤ ❞❡♠ ❣❡r❛❞❡ −1 ❣❡③❡✐❣t❡♥ ❣✐❜t ❡s j, k ♠✐t a = τ1 τj ✉♥❞ b = τ1 τk ✳ ❆✉s a = (τ1 τj )−1 = τj τ1 ❢♦❧❣t ❋ür ab = τj τ1 τ1 τk = τj τk ∈ F. ❛❧s♦ ❞✐❡ ❇❡❤❛✉♣t✉♥❣✳ ✸✵ ❆✉❢❣❛❜❡ ✹✳✽✳ ♥✐✉s❣r✉♣♣❡ G✳ ❙❡✐ H ❡✐♥ ❋r♦❜❡♥✐✉s❦♦♠♣❧❡♠❡♥t ❣❡r❛❞❡r ❖r❞♥✉♥❣ ✐♥ ❞❡r ❡♥❞❧✐❝❤❡♥ ❋r♦❜✲ ❩❡✐❣❡✱ ❞❛ss ❇❡♠❡r❦✉♥❣ ✹✳✾✳ ■st H H ❣❡♥❛✉ ❡✐♥ ❊❧❡♠❡♥t ❞❡r ❖r❞♥✉♥❣ 2 ❡♥t❤ä❧t✳ ❛✉✢ös❜❛r✱ ❞❛♥♥ ❧ässt s✐❝❤ ♠✐t ❞❡r s♦❣❡♥❛♥♥t❡♥ ❱❡r❧❛❣❡r✉♥❣s❛❜✲ ❜✐❧❞✉♥❣ ♦❤♥❡ ❈❤❛r❛❦t❡rt❤❡♦r✐❡ ❡❜❡♥❢❛❧❧s ③❡✐❣❡♥✱ ❞❛ss F ❡✐♥❡ ●r✉♣♣❡ ✐st✳ ●r✉♣♣❡♥ ✉♥❣❡r✲ ❛❞❡r ❖r❞♥✉♥❣ s✐♥❞ ③✇❛r ❛✉✢ös❜❛r ✭❋❡✐t✕❚❤♦♠♣s♦♥✮✱ ❛❜❡r ❞✐❡s❡r ❙❛t③ ✐st ❛✉❝❤ ❤❡✉t❡ ♥✉r s❡❤r ❛✉❢✇ä♥❞✐❣ ③✉ ❜❡✇❡✐s❡♥✱ ✉♥❞ ❜❡♥✉t③t ✈✐❡❧ ❈❤❛r❛❦t❡rt❤❡♦r✐❡✳ ❉❛❤❡r ❧✐❡❢❡rt ❞✐❡ ❡❧❡✲ |H| ♠❡♥t❛r❡ ❇❡❤❛♥❞❧✉♥❣ ❞❡r ❜❡✐❞❡♥ ❋ä❧❧❡ ❣❡r❛❞❡ ♦❞❡r H ❛✉✢ös❜❛r ♥♦❝❤ ❦❡✐♥ ❡✐♥❢❛❝❤❡s ❆r❣✉♠❡♥t ❢ür ❞❡♥ ❙❛t③ ✈♦♥ ❋r♦❜❡♥✐✉s✳ ❇❡♠❡r❦✉♥❣ ✹✳✶✵✳ Ü❜❡r ❞✐❡ ❙tr✉❦t✉r ❡♥❞❧✐❝❤❡r ❋r♦❜❡♥✐✉s❣r✉♣♣❡♥ ✐st ✈✐❡❧ ❜❡❦❛♥♥t✳ ❚❤♦♠♣s♦♥ ❜❡✇✐❡s ✐♥ s❡✐♥❡r ❉✐ss❡rt❛t✐♦♥ ❞✐❡ ❱❡r♠✉t✉♥❣ ✈♦♥ ❇✉r♥s✐❞❡✱ ❞❛ss ❞❡r ❋r♦❜❡✲ ♥✐✉s❦❡r♥ F ♥✐❧♣♦t❡♥t ✐st✱ ❛❧s♦ ❡✐♥ ❞✐r❡❦t❡s Pr♦❞✉❦t s❡✐♥❡r ❙②❧♦✇✉♥t❡r❣r✉♣♣❡♥ ✐st✳ ❆✉❝❤ |H| ✈♦♥ ✉♥❣❡r❛❞❡r ❖r❞♥✉♥❣✱ H ❡✐♥❡♥ ③②❦❧✐s❝❤❡♥ ◆♦r♠❛❧t❡✐❧❡r N ❤❛t✱ s♦ ❞❛ss ❛✉❝❤ H/N ③②❦❧✐s❝❤ ✐st✳ ❍❛t H ❣❡r❛❞❡ ❖r❞♥✉♥❣✱ ❞❛♥♥ ✐st Alt5 ❞❡r ❡✐♥③✐❣ ♠ö❣❧✐❝❤❡ ♥✐❝❤t ❛❜❡❧s❝❤❡ ❑♦♠♣♦s✐t✐♦♥s❢❛❦t♦r✳ ●❡♥❛✉❡r ❣✐❧t✿ ■st H ♥✐❝❤t (i) ❛✉✢ös❜❛r✱ ✉♥❞ i ♠✐t H = H (i+1) ✱ ❞❛♥♥ ❣✐❧t H (i) ∼ = SL2 (F5 )✳ ❞✐❡ ❙tr✉❦t✉r ❞❡s ❑♦♠♣❧❡♠❡♥ts ✉♥❞ ❞✐❡ ◆✐❧♣♦t❡♥③ ✈♦♥ F H ✐st s❡❤r ❡✐♥❣❡s❝❤rä♥❦t✿ ■st s❝❤♦♥ ❜❡❦❛♥♥t✱ ❞❛♥♥ ❦❛♥♥ ♠❛♥ ③❡✐❣❡♥✱ ❞❛ss ✺ ❈❤❛r❛❦t❡r❡ ✉♥❞ ●❛♥③❤❡✐t ✺✳✶ ●❛♥③ ❛❧❣❡❜r❛✐s❝❤❡ ❩❛❤❧❡♥ ❊✐♥✐❣❡ ✇✐❝❤t✐❣❡ ❆♥✇❡♥❞✉♥❣❡♥ ❞❡r ❈❤❛r❛❦t❡rt❤❡♦r✐❡ ❜❡r✉❤❡♥ ❛✉❢ ❡✐♥❡r ❱❡r❜✐♥❞✉♥❣ ③✉♠ ❇❡❣r✐✛ ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤❡r ❩❛❤❧❡♥✳ ❉❡✜♥✐t✐♦♥ ✺✳✶✳ ❊✐♥❡ ❦♦♠♣❧❡①❡ ❩❛❤❧ α ∈ C ❤❡✐ÿt ❣❛♥③ Z[X] ✐st✳ ♦❞❡r ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤ ✱ ✇❡♥♥ α ◆✉❧❧st❡❧❧❡ ❡✐♥❡s ♥♦r♠✐❡rt❡♥ P♦❧②♥♦♠s ❛✉s ❲✐r s❡❤❡♥ ❛❧s♦✱ ❞❛ss ❞✐❡ ●❛♥③❤❡✐t ❡✐♥❡ ❱❡rs❝❤är❢✉♥❣ ❞❡r ❆❧❣❡❜r❛✐③✐tät ✐st✳ ❱❡r✇❛♥❞t ♠✐t ❞❡♠ ❇❡❣r✐✛ ❡✐♥❡r ❡♥❞❧✐❝❤❡♥ ❑ör♣❡r❡r✇❡✐t❡r✉♥❣ ✐st ❞❡r ❇❡❣r✐✛ ❡✐♥❡r ❣❛♥③❡♥ ❘✐♥❣❡r✲ ✇❡✐t❡r✉♥❣✳ ❉❡✜♥✐t✐♦♥ ✺✳✷✳ ❡♥❞❧✐❝❤❡ ❊r✇❡✐t❡r✉♥❣ ✈♦♥ Z ♦❞❡r ❛✉❝❤ ❡♥❞❧✐❝❤ ü❜❡r Z✱ ✇❡♥♥ ❡s ❡♥❞❧✐❝❤ ✈✐❡❧❡ ❊❧❡♠❡♥t❡ r1 , r2 , .

N✱ ❊❧❡♠❡♥t❡ ❞❡r ❖r❞♥✉♥❣ 2✳ ❲✐r ❜❡❤❛✉♣t❡♥✱ ❞❛ss τi τj ∈ F ❢ür i, j ✳ ❉❛s ✐st ❦❧❛r ❢ür i = j ✳ ❙❡✐ ♥✉♥ i = j ✱ ✉♥❞ τi τj ∈ / F ✳ ❉❛♥♥ ❣✐❧t τi τj ∈ Hk ❢ür ❡✐♥ −1 k ✳ ◆❛tür❧✐❝❤ ❣✐❧t ❞❛♥♥ ❛✉❝❤ (τi τj ) ∈ Hk ✳ ❆♥❞❡r❡rs❡✐ts ❣✐❧t (τi τj )−1 = τj τi = (τi τj )τi ✱ ❙❡✐ ❛❧s♦ [G : H]✱ G ❡✐♥❡ ❡♥❞❧✐❝❤❡ ❋r♦❜❡♥✐✉s❣r✉♣♣❡ ♠✐t ❑♦♠♣❧❡♠❡♥t ✉♥❞ ❛❧s♦ e = (τi τj )−1 ∈ Hk ∩ Hkτi , ✉♥❞ ❞❛❤❡r τi ∈ Hk ✳ ❆❜❡r ❞❛♥♥ ❣✐❧t ❛✉❝❤ τj ∈ Hk ✱ ❛❧s♦ i = k = j✱ ✐♠ ❲✐❞❡rs♣r✉❝❤ ③✉ i = j✳ j = 1, 2, . . , n s✐♥❞ ❞✐❡ n ❊❧❡♠❡♥t❡ τ1 τj ♣❛❛r✇❡✐s❡ ✈❡rs❝❤✐❡❞❡♥✳ ❆❜❡r |F | = n✱ ❞❛❤❡r ❜❡st❡❤t F ❣❡♥❛✉ ❛✉s ❞✐❡s❡♥ ❊❧❡♠❡♥t❡♥✳ ❆❜❡r ❞❛♠✐t s❡❤❡♥ ✇✐r✱ ❞❛ss F ♠✉❧t✐♣❧✐❦❛t✐✈ −1 ❛❜❣❡s❝❤❧♦ss❡♥ ✐st✿ ❙❡✐❡♥ a, b ∈ F ✳ ❉❛♥♥ ❣✐❧t ❛✉❝❤ a ∈ F ✱ ✉♥❞ ♥❛❝❤ ❞❡♠ ❣❡r❛❞❡ −1 ❣❡③❡✐❣t❡♥ ❣✐❜t ❡s j, k ♠✐t a = τ1 τj ✉♥❞ b = τ1 τk ✳ ❆✉s a = (τ1 τj )−1 = τj τ1 ❢♦❧❣t ❋ür ab = τj τ1 τ1 τk = τj τk ∈ F.

G, h∈C ▲❡♠♠❛ ✺✳✼✳ ❙❡✐ g ∈ C ✳ ❉❛♥♥ ❣✐❧t ωχ (C) = χ(g)|C| . h✱ h ∈ C ✱ ✈♦♥ ϕC ❞✐❡ ❙♣✉r χ(g)✱ ❛❧s♦ ❣✐❧t ❉✐❡ ❇❡❤❛✉♣t✉♥❣ ❢♦❧❣t✳ χ(g)|C| ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤ χ(e) ♥❛❝❤ ❞❡♥ ♦❜✐❣❡♥ ❘❡s✉❧t❛t❡♥ ❯♥s❡r ♥ä❝❤st❡s ❩✐❡❧ ✐st ❡s ③✉ ③❡✐❣❡♥✱ ❞❛ss ❞✐❡ ❦♦♠♣❧❡①❡♥ ❩❛❤❧❡♥ χ(g) ❡✐♥❡ ❙✉♠♠❡ ✈♦♥ ❊✐♥❤❡✐ts✇✉r③❡❧♥ ✐st✱ ✐st χ(g) ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤✱ ✉♥❞ ❞❛♥♥ ✐st ❛✉❝❤ χ(g)|C| ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤✳ ❞✐❡ ❣❛♥③❡ ❩❛❤❧ χ(e) ❦ö♥♥t❡ ❞✐❡ ●❛♥③❤❡✐t ③❡rstör❡♥✳ s✐♥❞✳ ❉❛ ❆❜❡r ❞✐❡ ❉✐✈✐s✐♦♥ ❞✉r❝❤ ▲❡♠♠❛ ✺✳✽✳ ❙❡✐❡♥ C ✉♥❞ C ③✇❡✐ ✭♥✐❝❤t ♥♦t✇❡♥❞✐❣ ✈❡rs❝❤✐❡❞❡♥❡✮ ❑♦♥❥✉❣❛t✐♦♥s❦❧❛ss❡♥ ✈♦♥ G✳ ❙❡✐ ψ(g) ❞✐❡ ❆♥③❛❤❧ ❞❡r ❋❛❦t♦r✐s✐❡r✉♥❣❡♥ g = cc ♠✐t c ∈ C ✱ c ∈ C ✳ ❉❛♥♥ ✐st ψ ❡✐♥❡ ❑❧❛ss❡♥❢✉♥❦t✐♦♥✳ Pr♦♦❢✳ ❙❡✐❡♥ g ✉♥❞ h ❦♦♥❥✉❣✐❡rt✱ ❛❧s♦ h = g x ♠✐t x ∈ G✳ ❲❡❣❡♥ h = g x = (cc )x = cx c x x x ✐st (c, c ) → (c , c ) ❡✐♥❡ ❇✐❥❡❦t✐♦♥ ❞❡r P❛❛r❡ c, c ♠✐t g = cc ✉♥❞ ❞❡♥ ❡♥ts♣r❡❝❤❡♥❞❡♥ P❛❛r❡♥ ❢ür h✳ ▲❡♠♠❛ ✺✳✾✳ ❊s s❡✐❡♥ C1 , C2 , .

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