On finite groups isospectral to simple classical groups

TL;DR

This paper proves that finite groups sharing the same spectrum as large simple classical groups are structurally very similar, with restrictions on their composition factors and automorphism groups, extending understanding of spectral characterizations.

Contribution

It establishes that finite groups isospectral to large simple classical groups have highly constrained structures, particularly regarding their composition factors and automorphism relations.

Findings

Groups isospectral to large simple classical groups have restricted composition factors.

Such groups are closely related to the original simple group, often contained within its automorphism group.

The results limit the diversity of groups sharing the same spectrum as these classical groups.

Abstract

The spectrum $\omega(G)$ of a finite group $G$ is the set of element orders of $G$. Finite groups $G$ and $H$ are isospectral if their spectra coincide. Suppose that $L$ is a simple classical group of sufficiently large dimension (the lower bound varies for different types of groups but is at most 62) defined over a finite field of characteristic $p$. It is proved that a finite group $G$ isospectral to $L$ cannot have a nonabelian composition factor which is a group of Lie type defined over a field of characteristic distinct from $p$. Together with a series of previous results this implies that every finite group $G$ isospectral to $L$ is `close' to $L$. Namely, if $L$ is a linear or unitary group, then $L\leqslant G\leqslant\operatorname{Aut}(L)$, in particular, there are only finitely many such groups $G$ for given $L$. If $L$ is a symplectic or orthogonal group, then $G$ has a unique nonabelian composition factor $S$ and, for given $L$, there are at most 3 variants for $S$ (including $S\simeq L$).