Vol. 12, No. 6, 2018

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Bases for quasisimple linear groups

Melissa Lee and Martin W. Liebeck

Vol. 12 (2018), No. 6, 1537–1557
Abstract

Let V be a vector space of dimension d over Fq, a finite field of q elements, and let G GL(V )GLd(q) be a linear group. A base for G is a set of vectors whose pointwise stabilizer in G is trivial. We prove that if G is a quasisimple group (i.e., G is perfect and GZ(G) is simple) acting irreducibly on V , then excluding two natural families, G has a base of size at most 6. The two families consist of alternating groups Altm acting on the natural module of dimension d = m 1 or m 2, and classical groups with natural module of dimension d over subfields of Fq.

Keywords
linear groups, simple groups, representations, primitive permutation groups, bases of permutation groups
Mathematical Subject Classification 2010
Primary: 20C33
Secondary: 20B15, 20D06
Milestones
Received: 20 February 2018
Revised: 10 April 2018
Accepted: 6 June 2018
Published: 6 October 2018
Authors
Melissa Lee
Department of Mathematics
Imperial College
London
United Kingdom
Martin W. Liebeck
Department of Mathematics
Imperial College
London
United Kingdom