Involutions, weights and $p$-local structure

Abstract

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 $\left(Q,y\right)$, with a $p$-regular element $y$ and $Q$ a maximal $y$-invariant $p$-subgroup, give rise to $p$-blocks of defect zero of $N_G\left(Q\right)∕Q$, and we give lower bounds for the number of such blocks which arise. This relates to the weight conjecture of J. L. Alperin.