On the Prime Graph Question for Almost Simple Groups with an Alternating Socle

TL;DR

This paper investigates the Prime Graph Question for almost simple groups with alternating socles, establishing a criterion for units of order pq in their integral group rings and verifying the question for groups with n ≤ 17.

Contribution

It proves a new criterion linking units in the group ring to elements in the group for primes greater than n/3 and verifies the Prime Graph Question for all such groups with n ≤ 17.

Findings

Units of order pq correspond to group elements for primes > n/3.

Confirmed the Prime Graph Question for all almost simple groups with socle A_n, n ≤ 17.

Established a criterion connecting group ring units and group elements for large primes.

Abstract

Let $G$ be an almost simple group with socle $A_n$, the alternating group of degree $n$. We prove that there is a unit of order $pq$ in the integral group ring of $G$ if and only if there is an element of that order in $G$ provided $p$ and $q$ are primes greater than $\frac{n}{3}$. We combine this with some explicit computations to verify the Prime Graph Question for all almost simple groups with socle $A_n$ if $n \leq 17$.