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Darstellungstheorie endlicher Gruppen by Peter Müller

By Peter Müller

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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.

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.

St χ(e)2 − c2χ = (χ(e) − cχ )(χ(e) + cχ ) = |H|[1G , χ]H (χ(e) + cχ ) ❞✉r❝❤ H t❡✐❧❜❛r✱ ❞✳❤✳ ❞✐❡ ❧✐♥❦❡ ❙❡✐t❡ ✈♦♥ ✭✻✮ ✐st ❣❛♥③③❛❤❧✐❣✳ ❲✐r ❡r❤❛❧t❡♥ [χ, χ]H − ❆♥❞❡r❡rs❡✐ts ✐st [χ, χ]H c2χ χ(e)2 + ≤ 0. |H| |H| ❡✐♥❡ ❙✉♠♠❡ ♥✐❝❤t ♥❡❣❛t✐✈❡r ❙✉♠♠❛♥❞❡♥✱ ✇♦❜❡✐ χ(e)2 ❡✐♥ ❙✉♠✲ |H| ♠❛♥❞ ✐st✳ ❉❛❤❡r ❣✐❧t [χ, χ]H ≥ ❛♥❞ ✇✐r ❡r❤❛❧t❡♥ ❆✉❢❣❛❜❡ ✹✳✻✳ c2χ ≤ 0✱ ❙❡✐ G cχ = 0 ∈ N0 ✳ ❡✐♥❡ ●r✉♣♣❡ ♠✐t ■♥❦❧✉s✐♦♥✮ ❯♥t❡r❣r✉♣♣❡ A ❛❧s♦ A✳ ❩❡✐❣❡✿ ■st ❡✐♥ ❋r♦❜❡♥✐✉s❦♦♠♣❧❡♠❡♥t ✈♦♥ ❆✉❢❣❛❜❡ ✹✳✼✳ ❊s s❡✐ G χ(e)2 , |H| A |Z(G)| = 1✱ ✉♥❞ ❡✐♥❡r ♠❛①✐♠❛❧❡♥ ✭❜❡③ü❣❧✐❝❤ ❛❜❡❧s❝❤ ✉♥❞ ❦❡✐♥ ◆♦r♠❛❧t❡✐❧❡r ✈♦♥ G✱ ❞❛♥♥ ✐st G✳ ❡✐♥❡ ❡♥❞❧✐❝❤❡✱ ♥✐❝❤t ❛❜❡❧s❝❤❡✱ ❡✐♥❢❛❝❤❡ ●r✉♣♣❡✱ ✉♥❞ ♠❛①✐♠❛❧❡ ❯♥t❡r❣r✉♣♣❡ ✈♦♥ G✳ ❩❡✐❣❡✱ ❞❛ss H H ❡✐♥❡ ♥✐❝❤t ❛❜❡❧s❝❤ ✐st✳ ✹✳✸ ❋r♦❜❡♥✐✉s❦♦♠♣❧❡♠❡♥t ❣❡r❛❞❡r ❖r❞♥✉♥❣ ❲✐❡ s❝❤♦♥ ❡r✇ä❤♥t✱ ✐st ❡s ❜✐s ❤❡✉t❡ ♦❤♥❡ ❱❡r✇❡♥❞✉♥❣ ❞❡r ❈❤❛r❛❦t❡rt❤❡♦r✐❡ ♥✐❝❤t ♠ö❣❧✐❝❤ ③✉ ③❡✐❣❡♥✱ ❞❛ss ❋r♦❜❡♥✐✉s❦❡r♥❡ ✐♥ ❡♥❞❧✐❝❤❡♥ ❋r♦❜❡♥✐✉s❣r✉♣♣❡♥ ❯♥t❡r❣r✉♣♣❡♥ s✐♥❞✳ ❊✐♥ ❡✐♥❢❛❝❤❡r ❇❡✇❡✐s ✐st ❛❧❧❡r❞✐♥❣s ❢ür ❞❡♥ ❋❛❧❧ ❜❡❦❛♥♥t✱ ❞❛ss ❞❛s ❋r♦❜❡♥✐✉s❦♦♠♣❧❡♠❡♥t H ❣❡r❛❞❡ ❖r❞♥✉♥❣ ❤❛t✳ ❉❡r ❢♦❧❣❡♥❞❡ ❇❡✇❡✐s ✐st ❡✐♥❡ ✈♦♥ ❇❡♥❞❡r st❛♠♠❡♥❞❡ ❱❛r✐❛♥t❡ ❞❡s ✉rs♣rü♥❣❧✐❝❤❡♥ ❆r❣✉♠❡♥ts ✈♦♥ ❇✉r♥s✐❞❡✳ H ❣❡r❛❞❡r ❖r❞♥✉♥❣✱ n = F ❞❡r ❋r♦❜❡♥✐✉s❦❡r♥✳ ❙❡✐❡♥ H1 ✱ H2 ✱ ✳ ✳ ✳ ✱Hn ❞✐❡ n ✈❡rs❝❤✐❡❞❡♥❡♥ ❑♦♥✲ ❥✉❣✐❡rt❡♥ ✈♦♥ H ✳ ❊s ❣✐❧t ❛❧s♦ Hi ∩ Hj = {e} ❢ür i = j ✱ ✉♥❞ |F | = n✳ ❙❡✐❡♥ τi ∈ Hi ✱ i = 1, 2, .

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