Quasisimple classical groups and their complex group algebras

TL;DR

This paper demonstrates that, with few exceptions, finite quasisimple classical groups are uniquely identified by their complex group algebras, leveraging classification theorems and character degree analyses.

Contribution

It establishes the uniqueness of quasisimple classical groups from their complex group algebras, except for certain cases with exceptional Schur multipliers.

Findings

Most quasisimple classical groups are uniquely determined by their complex group algebras.

The classification of finite simple groups is crucial for the proof.

Exceptional Schur multipliers are the main known exceptions.

Abstract

Let $H$ be a finite quasisimple classical group, i.e. $H$ is perfect and $S:=H/Z(H)$ is a finite simple classical group. We prove in this paper that, excluding the cases when the simple group $S$ has a very exceptional Schur multiplier such as $\PSL_3(4)$ or $\PSU_4(3)$, $H$ is uniquely determined by the structure of its complex group algebra. The proofs make essential use of the classification of finite simple groups as well as the results on prime power character degrees and relatively small character degrees of quasisimple classical groups.