On the Prime Graph Question for Integral Group Rings of Conway simple groups

TL;DR

This paper addresses the Prime Graph Question for the integral group rings of Conway sporadic simple groups, linking it to combinatorial problems involving Young tableaux and Littlewood-Richardson coefficients, thus completing prior research.

Contribution

It proves the Prime Graph Question for the Conway sporadic simple groups by reducing it to combinatorial problems, finishing previous work in the area.

Findings

Confirmed the Prime Graph Question for the three Conway sporadic groups.

Reduced the problem to combinatorial questions about Young tableaux.

Connected group ring properties to Littlewood-Richardson coefficients.

Abstract

The Prime Graph Question for integral group rings asks if it is true that if the normalized unit group of the integral group ring of a finite group $G$ contains an element of order $pq$, for some primes $p$ and $q$, also $G$ contains an element of that order. We answer this question for the three Conway sporadic simple groups after reducing it to a combinatorial question about Young tableaus and Littlewood-Richardson coefficients. This finishes work of V. Bovdi, A. Konovalov and S. Linton.