Base sizes of primitive permutation groups

2021/06/02 by Moscatiello, Mariapia, Roney-Dougal, Colva M. · 2 citations
#20B10 #20B15 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2106.01219

Abstract

Let G be a permutation group, acting on a set Ωof size n. A subset B of Ωis a base for G if the pointwise stabilizer G_(B) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of Sym(n) is large base if there exist integers m and r ≥ 1 such that Alt(m)^r \unlhd G ≤ Sym(m) \wr Sym(r), where the action of Sym(m) is on k-element subsets of 1,...,m and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group M24 in its natural action on 24 points, or b(G) ≤ \lceil log n\rceil+1. Furthermore, we show that there are infinitely many primitive groups G that are not large base for which b(G) > log n + 1, so our bound is optimal.

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• On the base size and minimal degree of transitive groups
• The Saxl hypergraph of a permutation group

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