On orders of elements of finite almost simple groups with linear or unitary socle

TL;DR

This paper characterizes admissible almost simple groups with socle a finite simple linear or unitary group, focusing on element orders and automorphisms, providing a detailed description of their spectra.

Contribution

It identifies all admissible almost simple groups with socle a finite simple linear or unitary group and computes element orders in specific cosets involving automorphisms.

Findings

Characterization of admissible almost simple groups with linear or unitary socle

Explicit calculation of element orders in cosets involving automorphisms

Description of spectra for these groups and cosets

Abstract

We say that a finite almost simple $G$ with socle $S$ is admissible (with respect to the spectrum) if $G$ and $S$ have the same sets of orders of elements. Let $L$ be a finite simple linear or unitary group of dimension at least three over a field of odd characteristic. We describe admissible almost simple groups with socle $L$. Also we calculate the orders of elements of the coset $L\tau$, where $\tau$ is the inverse-transpose automorphism of $L$.