An application of blocks to torsion units in group rings

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.