A general Brauer-Fowler theorem and centralizers in locally finite groups

Brian Hartley

Vol. 152 (1992), No. 1, 101–117

DOI: 10.2140/pjm.1992.152.101

Abstract

A classical theorem of Brauer and Fowler, proved by quite elementary arguments, states that the order of a finite non-abelian simple group G is bounded in terms of the order of the centralizer of any involution in G. Here we use the classification of finite simple groups to show that the order of such a G is bounded in terms of the order of any automorphism α and the number of fixed points of α. It follows easily that if a locally finite group contains an element with finite centralizer, then it has a locally solvable subgroup of finite index.

Mathematical Subject Classification 2000

Primary: 20F28

Milestones

Received: 30 August 1990

Revised: 6 March 1991

Published: 1 January 1992