On the prime graph of a finite group with unique nonabelian composition factor

TL;DR

This paper investigates the structure of finite groups with a unique nonabelian composition factor, showing that their almost simple quotient is a cyclic extension of its socle, and explores prime graph properties related to these groups.

Contribution

It proves that the almost simple group associated with such finite groups is a cyclic extension of its socle, and refines existing results on prime graphs and recognizability.

Findings

The almost simple group is a cyclic extension of its socle.

Conditions are established for the prime graph to have a universal vertex.

Refinement of results on finite groups almost recognizable by prime graph.

Abstract

We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group $G$ isospectral to a finite simple group has a unique nonabelian composition factor, that is, the quotient of $G$ by the solvable radical of $G$ is an almost simple group. The main goal of this paper is prove that this almost simple group is a cyclic extension of its socle. To this end, we consider a general situation when $G$ is an arbitrary group with unique nonabelian composition factor, not necessarily isospectral to a simple group, and study the prime graph of $G$, where the prime graph of $G$ is the graph whose vertices are the prime numbers dividing the order of $G$ and two such numbers $r$ and $s$ are adjacent if and only if $r\neq s$ and $G$ has an element of order $rs$. Namely, we establish some sufficient conditions for the prime graph of such a group to have a vertex adjacent to all other vertices. Besides proving the main result, this allows us to refine a recent result by P. Cameron and N. Maslova concerning finite groups almost recognizable by prime graph.