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On the minimal degree and base size of finite primitive groups

Fabio Mastrogiacomo

Abstract

Let $G$ be a finite permutation group acting on $Ω$. A base for $G$ is a subset $B \subseteq Ω$ such that the pointwise stabilizer $G_{(B)}$ is the identity. The base size of $G$, denoted by $b(G)$, is the cardinality of the smallest possible base. The minimal degree of $G$, denoted by $μ(G)$, is the smallest cardinality of the support of a non trivial element of $G$. In this paper, we establish a new upper bound for $b(G)$ when $G$ is primitive, and subsequently prove that if $G$ is a primitive group different from the Mathieu group of degree $24$, then $μ(G)b(G)\leq n \log n$, where $n$ is the degree of $G$. This bound is best possible, up to a multiplicative constant.

On the minimal degree and base size of finite primitive groups

Abstract

Let be a finite permutation group acting on . A base for is a subset such that the pointwise stabilizer is the identity. The base size of , denoted by , is the cardinality of the smallest possible base. The minimal degree of , denoted by , is the smallest cardinality of the support of a non trivial element of . In this paper, we establish a new upper bound for when is primitive, and subsequently prove that if is a primitive group different from the Mathieu group of degree , then , where is the degree of . This bound is best possible, up to a multiplicative constant.

Paper Structure

This paper contains 16 sections, 37 theorems, 153 equations, 2 tables.

Table of Contents

  1. introduction
  2. Proof of Theorem \ref{['mainthm:2']}
  3. Almost simple groups
  4. The socle of $G$ is an alternating group
  5. The socle of $G$ is a classical group
  6. Diagonal groups
  7. Affine groups
  8. Product action
  9. Proof of Theorem \ref{['mainthm:1']}
  10. Affine groups
  11. Almost simple groups as in Table \ref{['tab:md']}
  12. Linear groups
  13. Symplectic groups
  14. Orthogonal groups of even dimension
  15. Orthogonal groups in odd dimension
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $G$ be a primitive permutation group of degree $n$. If $G$ is not the Mathieu group $M_{24}$ in its action of degree $24$, then

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • ...and 61 more