On finite groups isospectral to groups with abelian Sylow $2$-subgroups

TL;DR

This paper investigates which finite groups are uniquely determined by their element order spectrum, especially focusing on products of simple groups with abelian Sylow 2-subgroups and certain sporadic groups.

Contribution

It identifies specific nonabelian simple groups whose direct products are uniquely determined by their spectra and shows infinite families of groups sharing spectra with certain Ree and J1 groups.

Findings

Direct products of certain simple groups are uniquely identified by their spectra.

Infinitely many groups share spectra with the cube of the Ree group and the fourth power of J1.

Abstract

The spectrum of a finite group is the set of orders of its elements. We are concerned with finite groups having the same spectrum as a direct product of nonabelian simple groups with abelian Sylow $2$-subgroups. For every positive integer $k$, we find $k$ nonabelian simple groups with abelian Sylow 2-subgroups such that their direct product is uniquely determined by its spectrum in the class of all finite groups. On the other hand, we prove that there are infinitely many finite groups having the same spectrum as the direct cube of the small Ree group $^2G_2(q)$, $q>3$, or the direct fourth power of the sporadic group $J_1$.