Complex group algebras of the double covers of the symmetric and alternating groups

TL;DR

This paper proves that the complex group algebra uniquely determines the double covers of symmetric and alternating groups, confirming a conjecture that finite quasi-simple groups are distinguished by their algebraic structure.

Contribution

It establishes that the complex group algebra characterizes the double covers of symmetric and alternating groups, completing a conjecture about finite quasi-simple groups.

Findings

Double covers of symmetric and alternating groups are uniquely determined by their complex group algebras.

The result confirms that finite quasi-simple groups are uniquely identified by their algebraic structure.

The proof relies on known representation degrees and the classification of finite simple groups.

Abstract

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let $n\geq 5$ be an integer, $G$ a finite group, and let $\AAA$ and $\SSS^\pm$ denote the double covers of $\Al_n$ and $\Sy_n$, respectively. We prove that $\CC G\cong \CC \AAA$ if and only if $G\cong \AAA$, and $\CC G\cong \CC \SSS^+\cong\CC\SSS^-$ if and only if $G\cong \SSS^+$ or $\SSS^-$. This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasi-simple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.