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On the depth of subgroups of simple groups

Timothy C. Burness

Abstract

The depth of a subgroup $H$ of a finite group $G$ is a positive integer defined with respect to the inclusion of the corresponding complex group algebras $\mathbb{C}H \subseteq \mathbb{C}G$. This notion was originally introduced by Boltje, Danz and Külshammer in 2011, and it has been the subject of numerous papers in recent years. In this paper, we study the depth of core-free subgroups, which allows us to apply powerful computational and probabilistic techniques that were originally designed for studying bases for permutation groups. We use these methods to prove a wide range of new results on the depth of subgroups of almost simple groups, significantly extending the scope of earlier work in this direction. For example, we establish best possible bounds on the depth of irreducible subgroups of classical groups and primitive subgroups of symmetric groups. And with the exception of a handful of open cases involving the Baby Monster, we calculate the exact depth of every subgroup of every almost simple sporadic group. We also present a number of open problems and conjectures.

On the depth of subgroups of simple groups

Abstract

The depth of a subgroup of a finite group is a positive integer defined with respect to the inclusion of the corresponding complex group algebras . This notion was originally introduced by Boltje, Danz and Külshammer in 2011, and it has been the subject of numerous papers in recent years. In this paper, we study the depth of core-free subgroups, which allows us to apply powerful computational and probabilistic techniques that were originally designed for studying bases for permutation groups. We use these methods to prove a wide range of new results on the depth of subgroups of almost simple groups, significantly extending the scope of earlier work in this direction. For example, we establish best possible bounds on the depth of irreducible subgroups of classical groups and primitive subgroups of symmetric groups. And with the exception of a handful of open cases involving the Baby Monster, we calculate the exact depth of every subgroup of every almost simple sporadic group. We also present a number of open problems and conjectures.

Paper Structure

This paper contains 23 sections, 22 theorems, 61 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Depth
  4. The inclusion matrix
  5. The Külshammer graph
  6. Bases
  7. Almost simple groups
  8. Computational methods
  9. Magma computations
  10. GAP computations
  11. Proof of Theorems \ref{['t:ns']}-\ref{['t:class']}
  12. Proof of Theorem \ref{['t:ns']}
  13. Proof of Theorem \ref{['t:sn']}
  14. Proof of Theorem \ref{['t:sn2']}
  15. Proof of Theorem \ref{['t:class']}
  16. ...and 8 more sections

Key Result

Theorem 1

Let $G \leqslant {\rm Sym}(\Omega)$ be a non-standard almost simple primitive permutation group with socle $G_0$ and point stabilizer $H$.

Theorems & Definitions (59)

  • Definition
  • Theorem 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Theorem 9
  • ...and 49 more