Document Type : Research Paper

**Author**

**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 $L\leq G\leq Aut L$. We show that the assertion remains true if 62 is replaced by 38.

**Keywords**

- Simple classical groups
- Element orders
- Prime graph of a finite group
- Almost recognizable group

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December 2017

Pages 7-33

**Receive Date:**11 July 2016**Revise Date:**05 December 2016**Accept Date:**11 December 2016**Published Online:**01 December 2017