Base sizes for primitive groups with soluble stabilisers

Tim C Burness*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

16 Citations (Scopus)
83 Downloads (Pure)

Abstract

Let G be a finite primitive permutation group on a set  with point stabiliser H. Recall that a subset of is a base for G if its pointwise stabiliser is trivial. We define the base size of G, denoted b(G, H), to be the minimal size of a base for G. Determining the base size of a group is a fundamental problem in permutation group theory, with a long history stretching back to the 19th century. Here one of our main motivations is a theorem of Seress from 1996, which states that b(G, H) 6 4 if G is soluble. In this paper we extend Seress’s result by proving that b(G, H) 6 5 for all finite primitive groups G with a soluble point stabiliser H. This bound is best possible. We also determine the exact base size for all almost
simple groups and we study random bases in this setting. For example, we prove that the probability that 4 random elements in  form a base tends to 1 as |G| tends to infinity.
Original languageEnglish
Pages (from-to)1755-1807
Number of pages56
JournalAlgebra and Number Theory
Volume15
Issue number7
DOIs
Publication statusPublished - 1 Nov 2021

Bibliographical note

Publisher Copyright:
© 2021 Mathematical Sciences Publishers.

Keywords

  • Primitive groups
  • base sizes
  • soluble stabilisers

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