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

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.