Bases for quasisimple linear groups

Lee, Melissa; Liebeck, Martin

doi:10.2140/ant.2018.12.1537

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Bases for quasisimple linear groups

Melissa Lee and Martin W. Liebeck

Vol. 12 (2018), No. 6, 1537–1557

DOI: 10.2140/ant.2018.12.1537

Abstract

Let $V$ be a vector space of dimension $d$ over $F_{q}$ , a finite field of $q$ elements, and let $G \leq GL (V) ≅ {GL}_{d} (q)$ 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 $G ∕ Z (G)$ 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 ${Alt}_{m}$ 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 $F_{q}$ .

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

Vol. 12, No. 6, 2018