An application of blocks to torsion units in group rings

TL;DR

This paper applies block theory of cyclic defect to analyze the existence of elements of order pq in the normalized unit group of integral group rings, confirming the Prime Graph Question for certain groups.

Contribution

It introduces a novel application of block theory to relate units in group rings to group elements, verifying the Prime Graph Question for specific classes of groups.

Findings

Confirmed the Prime Graph Question for all alternating and symmetric groups.

Verified the Prime Graph Question for two sporadic simple groups.

Established a criterion linking units of order pq to group elements under certain block conditions.

Abstract

We use the theory of blocks of cyclic defect to prove that under a certain condition on the principal p-block of a finite group G the normalized unit group of the integral group ring of G contains an element of order pq if and only if so does G, for q a prime different from p. Using this we verify the Prime Graph Question for all alternating and symmetric groups and also for two sporadic simple groups.