# Groups, Geometry, and Dynamics

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**Volume 1, Issue 4, 2007, pp. 347–400**

**DOI: 10.4171/GGD/18**

Published online: 2007-12-31

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

Michael Aschbacher^{[1]}, Robert M. Guralnick

^{[2]}and Yoav Segev

^{[3]}(1) California Institute of Technology, Pasadena, USA

(2) University of Southern California, Los Angeles, United States

(3) Ben-Gurion University, Beer-Sheva, Israel

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.

*Keywords: *Finite simple groups, involutions, parabolic subgroups, fundamental subgroups, saturation

Aschbacher Michael, Guralnick Robert, Segev Yoav: Elementary abelian 2-subgroups of Sidki-type in finite groups. *Groups Geom. Dyn.* 1 (2007), 347-400. doi: 10.4171/GGD/18