# Bases of primitive linear groups

@article{Liebeck2002BasesOP,
title={Bases of primitive linear groups},
author={Martin W. Liebeck and Aner Shalev},
journal={Journal of Algebra},
year={2002},
volume={252},
pages={95-113}
}
• Published 1 June 2002
• Mathematics
• Journal of Algebra
26 Citations
• Mathematics
• 2002
Let G be a permutation group on a finite set Ω of size n. A subset of Ω is said to be a base for G if its pointwise stabilizer in G is trivial. The minimal size of a base for G is denoted by b(G).
• Mathematics
• 2010
We give a description of infinite families of finite primitive permutation groups for which there is a uniform finite upper bound on the diameter of all orbital graphs. This is equivalent to
• Mathematics
• 2008
The following article appeared in Annals of Mathematics 173.2 (2011): 769-814 and may be found at http://annals.math.princeton.edu/2011/173-2/p04
• Mathematics
• 2013
Let G be a permutation group on a finite set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. The base size of G, denoted b(G), is the smallest size of a base. A
• Mathematics
Bulletin of the London Mathematical Society
• 2019
For a finite group G , let diam (G) denote the maximum diameter of a connected Cayley graph of G . A well‐known conjecture of Babai states that diam (G) is bounded by (log2|G|)O(1) in case G is a
• Mathematics
• 2015
Let $V$ be a finite vector space over a finite field of order $q$ and of characteristic $p$. Let $G\leq GL(V)$ be a $p$-solvable completely reducible linear group. Then there exists a base for $G$ on

## References

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A base of a permutation group G is a sequence B of points from the permutation domain such that only the identity of G fixes B pointwise. Answering a question of Pyber, we prove that all primitive
• Mathematics
• 1998
Abstract A base of a permutation group G is a sequence B of points from the permutation domain such that only the identity of G fixes B pointwise. We show that primitive permutation groups with no
The author has determined, for all simple simply connected reductive linear algebraic groups defined over a finite field, all the irreducible representations in their defining characteristic of
• Mathematics
• 1983
Part I: Properties of $K$-groups and Preliminary Lemmas: Introduction Decorations of the known simple groups Local subgroups of the known simple groups Balance and signalizers Generational properties
• Mathematics
• 1998
Let G be a permutation group on a finite set Ω. A sequence B=(ω1, …, ωb) of points in Ω is called a base if its pointwise stabilizer in G is the identity. Bases are of fundamental importance in
• Mathematics
• 2001
Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph r(G, S) associated with G and S, up to a multiplicative constant. Many applications
• Mathematics
• 2000
Abstract If G is a group, H a subgroup of G , and Ω a transitive G -set we ask under what conditions one can guarantee that H has a regular orbit ( = of size | H |) on Ω. Here we prove that if PSL (
• Mathematics
• 1999
In recent years probabilistic methods have proved useful in the solution of several problems concerning finite groups, mainly involving simple groups and permutation groups. In some cases the
(1.1) Definition Let 1 6= G be a group. A subgroup M of G is said to be maximal if M 6= G and there exists no subgroup H such that M < H < G. IfG is finite, by order reasons every subgroupH 6= G is