Involutions, weights and p-local structure

Robinson, Geoffrey

doi:10.2140/ant.2011.5.1063

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Involutions, weights and $p$-local structure

Geoffrey R. Robinson

Vol. 5 (2011), No. 8, 1063–1068

DOI: 10.2140/ant.2011.5.1063

Abstract

We prove that for an odd prime $p$ , a finite group $G$ with no element of order $2 p$ 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 $N_{G} (Q) ∕ Q$ , and we give lower bounds for the number of such blocks which arise. This relates to the weight conjecture of J. L. Alperin.

Keywords

block, involution

Mathematical Subject Classification 2010

Primary: 20C20

Milestones

Received: 9 June 2010

Revised: 22 December 2010

Accepted: 7 June 2011

Published: 5 June 2012

Authors

Geoffrey R. Robinson
Institute of Mathematics University of Aberdeen Fraser Noble Building Aberdeen AB24 3UE Scotland

Vol. 5, No. 8, 2011