Fixed point ratios in actions of finite exceptional groups of Lie type

Ross Lawther, Martin W. Liebeck and Gary M. Seitz

Vol. 205 (2002), No. 2, 393–464

DOI: 10.2140/pjm.2002.205.393

Abstract

Let G be a finite exceptional group of Lie type acting transitively on a set . For x ∈ G, the fixed point ratio of x is the proportion of elements of which are fixed by x. We obtain new bounds for such fixed point ratios. When a point-stabilizer is parabolic we use character theory; and in other cases, we use results on an analogous problem for algebraic groups in Lawther, Liebeck & Seitz, 2002. These give dimension bounds on fixed point spaces of elements of exceptional algebraic groups, which we apply by passing to finite groups via a Frobenius morphism.

Milestones

Received: 10 October 2000

Revised: 26 September 2001

Published: 1 August 2002