On groups having the prime graph as alternating and symmetric groups

TL;DR

This paper characterizes finite groups whose prime graphs match those of large alternating or symmetric groups, showing they are closely related to these groups with a specific normal subgroup structure.

Contribution

It proves that such groups are essentially extensions of an alternating or symmetric group by a normal subgroup with limited prime divisors.

Findings

Groups with prime graph equal to that of A_n or S_n are closely related to these groups.

The structure involves a normal subgroup with at most one large prime divisor.

The result applies for all n ≥ 19.

Abstract

The {\it prime graph} $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of $G$ of order $rs$. Let $A_n$ ($S_n$) denote the alternating (symmetric) group of degree $n$. We prove that if $G$ is a finite group with $\Gamma(G)=\Gamma(A_n)$ or $\Gamma(G)=\Gamma(S_n)$, where $n\geq19$, then there exists a normal subgroup $K$ of $G$ and an integer $t$ such that $A_t\leq G/K\leq S_t$ and $|K|$ is divisible by at most one prime greater than $n/2$.