On the Prime Graph Question for Integral Group Rings of 4-primary groups II

TL;DR

This paper advances the understanding of the Prime Graph Question for integral group rings of almost simple groups with four prime divisors, introducing a new lattice method involving Littlewood-Richardson coefficients, and proving the Zassenhaus Conjecture for additional simple groups.

Contribution

It develops and applies the lattice method to infinite series of groups, providing new results and confirming the Zassenhaus Conjecture for four more simple groups.

Findings

Lattice method complements HeLP-method results

Proves Zassenhaus Conjecture for four simple groups

Positive answer to Prime Graph Question around vertex 3 with Sylow 3-subgroup of order 3

Abstract

In this article the study of the Prime Graph Question for the integral group ring of almost simple groups which have an order divisible by exactly $4$ different primes is continued. We provide more details on the recently developed "lattice method" which involves the calculation of Littlewood-Richardson coefficients. We apply the method obtaining results complementary to those previously obtained using the HeLP-method. In particular the "lattice method" is applied to infinite series of groups for the first time. We also prove the Zassenhaus Conjecture for four more simple groups. Furthermore we show that the Prime Graph Question has a positive answer around the vertex $3$ provided the Sylow $3$-subgroup is of order $3$.