Elementary abelian 2-subgroups of Sidki-type in finite groups

Abstract

Let $G$ be a finite group. We say that a nontrivial elementary abelian 2-subgroup $V$ of $G$ is of Sidki-type in $G$, if for each involution $i$ in $G$, $C_V(i)\ne 1$. A conjecture due to S. Sidki (J. Algebra 39, 1976) asserts that if $V$ is of Sidki-type in $G$, then $V\cap O_2(G)\ne 1$. In this paper we prove a stronger version of Sidki's conjecture. As part of the proof, we also establish weak versions of the saturation results of G. Seitz (Invent. Math. 141, 2000) for involutions in finite groups of Lie type in characteristic $2$. Seitz's results apply to elements of order $p$ in groups of Lie type in characteristic $p$, but only when $p$ is a good prime, and $2$ is usually not a good prime.