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Simple classical groups of Lie type are determined by their character degrees. (English) Zbl 1259.20008
Working over $$\mathbb C$$, let $$G$$ be a finite group and denote by $$\mathrm{Irr}(G)$$ the set of all irreducible (complex) characters of $$G$$. Let $$X(G)=\{\chi(1):\chi\in\mathrm{Irr}(G)\}$$ be the set of all irreducible character degrees, ignoring multiplicities. Let $$X_1(G)$$ be the degree pattern, namely the set of all irreducible character degrees of $$G$$, counting multiplicities (this is precisely the first column of the (ordinary) character table for $$G$$).
It is known by Molien’s theorem that determining the structure of the (complex) group algebra is equivalent to knowing $$X_1(G)$$. It was conjectured that all non-Abelian simple groups are uniquely determined by the structure of their group algebras, and subsequently this conjecture was verified for several classes of such groups. Huppert made a stronger (and still open) conjecture that, for $$H$$ any non-Abelian simple group such that $$X(G)=X(H)$$, then $$G\cong H\times A$$ where $$A$$ is Abelian.
The author proves that for $$H$$ a non-Abelian simple group and $$G$$ any finite group, if $$X_1(G)=X_1(H)$$, then $$G\cong H$$. He proves this result only for simple classical groups of Lie type, because the author has already shown the result to hold for all other non-Abelian simple groups. As a corollary therefore all non-Abelian simple groups are uniquely determined by the structure of their (complex) group algebras. The main lemma, which requires the classification of finite simple groups, considers non-Abelian simple groups of the form $$G/M$$ where $$G$$ is perfect and $$M$$ is a maximal normal subgroup.

##### MSC:
 20C15 Ordinary representations and characters 20C33 Representations of finite groups of Lie type 20D06 Simple groups: alternating groups and groups of Lie type 20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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