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

### Michael Aschbacher

California Institute of Technology, Pasadena, USA### Robert M. Guralnick

University of Southern California, Los Angeles, United States### Yoav Segev

Ben-Gurion University, Beer-Sheva, Israel

## 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 $G$, $C_{V}(i) =1$. A conjecture due to S. Sidki (J. Algebra 39, 1976) asserts that if $V$ is of Sidki-type in $G$, then $V ∩O_{2}(G) =1$. 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.

## Cite this article

Michael Aschbacher, Robert M. Guralnick, Yoav Segev, Elementary abelian 2-subgroups of Sidki-type in finite groups. Groups Geom. Dyn. 1 (2007), no. 4, pp. 347–400

DOI 10.4171/GGD/18