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Irredundant bases for the symmetric group

Colva M. Roney-Dougal, Peiran Wu

Abstract

An irredundant base of a group $G$ acting faithfully on a finite set $Γ$ is a sequence of points in $Γ$ that produces a strictly descending chain of pointwise stabiliser subgroups in $G$, terminating at the trivial subgroup. Suppose that $G$ is $\operatorname{S}_n$ or $\operatorname{A}_n$ acting primitively on $Γ$, and that the point stabiliser is primitive in its natural action on $n$ points. We prove that the maximum size of an irredundant base of $G$ is $O\left(\sqrt{n}\right)$, and in most cases $O\left((\log n)^2\right)$. We also show that these bounds are best possible.

Irredundant bases for the symmetric group

Abstract

An irredundant base of a group acting faithfully on a finite set is a sequence of points in that produces a strictly descending chain of pointwise stabiliser subgroups in , terminating at the trivial subgroup. Suppose that is or acting primitively on , and that the point stabiliser is primitive in its natural action on points. We prove that the maximum size of an irredundant base of is , and in most cases . We also show that these bounds are best possible.
Paper Structure (9 sections, 19 theorems, 62 equations)

This paper contains 9 sections, 19 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. The maximum irredundant base size
  3. The affine case
  4. Subspace stabilisers and the diagonal subgroup
  5. Subgroups of the diagonal subgroup
  6. Proof of Theorem \ref{['theorem:affine-case']}
  7. The product action case
  8. Proof of Theorem \ref{['theorem:optimal-upper-bounds']}
  9. Acknowledgement

Key Result

Theorem 1

Suppose $G$ is $\operatorname{S}_{n}$ or $\operatorname{A}_{n}$ ($n \geqslant 7$) and $H \neq \operatorname{A}_{n}$ is a primitive maximal subgroup of $G$.

Theorems & Definitions (36)

  • Theorem 1
  • Corollary 2
  • Remark 3
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 26 more