On recognition of symplectic and orthogonal groups of small dimensions by spectrum

TL;DR

This paper characterizes finite groups with the same element order spectrum as certain small-dimensional symplectic and orthogonal groups, showing most are extensions of these groups by automorphisms.

Contribution

It determines all finite groups isospectral to specific simple groups, revealing their structure as mostly automorphism extensions.

Findings

Most isospectral groups are automorphism extensions of the simple groups.

Four exceptions where the structure differs.

Complete classification of groups with matching spectra.

Abstract

We refer to the set of the orders of elements of a finite group as its spectrum and say that finite groups are isospectral if their spectra coincide. In the paper we determine all finite groups isospectral to the simple groups $S_6(q)$, $O_7(q)$, and $O_8^+(q)$. In particular, we prove that with just four exceptions, every such a finite group is an extension of the initial simple group by a (possibly trivial) field automorphism.