#### 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 ${\mathbb{F}}_{q}$, a finite field of $q$ elements, and let $G\le GL\left(V\right)\cong {GL}_{d}\left(q\right)$ 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 $G∕Z\left(G\right)$ 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 ${\mathbb{F}}_{q}$.

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##### 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