On almost recognizability by spectrum of simple classical groups

Abstract

‎The set of element orders of a finite group $G$ is called the {em spectrum}‎. ‎Groups with coinciding spectra are said to be {em isospectral}‎. ‎It is known that if $G$ has a nontrivial normal soluble subgroup then there exist infinitely many pairwise non-isomorphic‎ ‎groups isospectral to $G$‎. ‎The situation is quite different if $G$ is a nonabelain simple group‎. ‎Recently it was proved that if $L$ is a simple classical group of dimension at least 62 and $G$ is a finite group‎ ‎isospectral to $L$‎, ‎then up to isomorphism $Lleq GleqAut L$‎. ‎We show that the assertion remains true‎ ‎if 62 is replaced by 38‎.