Vol. 5, No. 8, 2011

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Involutions, weights and $p$-local structure

Geoffrey R. Robinson

Vol. 5 (2011), No. 8, 1063–1068

We prove that for an odd prime p, a finite group G with no element of order 2p has a p-block of defect zero if it has a non-Abelian Sylow p-subgroup or more than one conjugacy class of involutions. For p = 2, we prove similar results using elements of order 3 in place of involutions. We also illustrate (for an arbitrary prime p) that certain pairs (Q,y), with a p-regular element y and Q a maximal y-invariant p-subgroup, give rise to p-blocks of defect zero of NG(Q)Q, and we give lower bounds for the number of such blocks which arise. This relates to the weight conjecture of J. L. Alperin.

block, involution
Mathematical Subject Classification 2010
Primary: 20C20
Received: 9 June 2010
Revised: 22 December 2010
Accepted: 7 June 2011
Published: 5 June 2012
Geoffrey R. Robinson
Institute of Mathematics
University of Aberdeen
Fraser Noble Building
Aberdeen AB24 3UE