Simple classical groups of Lie type are determined by their character degrees.

*(English)*Zbl 1259.20008Working 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.

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.

Reviewer: Stuart Martin (Cambridge)

##### 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) |

##### Keywords:

character degrees; simple classical groups; complex group algebras; simple groups of Lie type; irreducible complex characters; isomorphism problem**OpenURL**

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