Properties of element orders in covers for L(n,q) and U(n,q)

TL;DR

This paper investigates the element orders in covers of groups like PSL(n,q) and PSU(n,q), showing that certain semidirect products contain elements with orders not present in the original groups, aiding in their recognition by spectrum.

Contribution

It demonstrates that under specific conditions, covers of PSL(n,q) and PSU(n,q) contain elements with unique orders, enabling recognition of these groups by their spectrum.

Findings

Semidirect products contain elements with unique orders.

PSL(n,q) is recognizable by spectrum from its covers under certain conditions.

Partial solution to recognition problem in Kourovka notebook.

Abstract

We show that if a finite simple group G isomorphic to PSL(n,q) or PSU(n,q), where either $n\ne 4$, or q is prime or even, acts on a vector space over a field of the defining characteristic of G, then the corresponding semidirect product contains an element whose order is distinct from every element order of G. As a consequence, we prove that the group PSL(n,q), where $n\ne 4$ or q prime or even, is recognizable by spectrum from its covers thus giving a partial positive answer to Problem 14.60 from the Kourovka notebook.