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Subnormalisers of semisimple elements in finite groups of Lie type

Gunter Malle

TL;DR

This work provides a comprehensive classification of subnormalisers of semisimple p-elements in finite quasi-simple groups of Lie type. It reduces the problem to understanding maximal overgroups of normalisers of Sylow d-tori, and then describes the subnormalisers explicitly in both exceptional and classical groups, using Aschbacher’s framework and Levi-subgroup decompositions. The authors extend the analysis to symmetric and sporadic groups to yield a complete landscape, with a notable algebraic-group construction linking finite subnormalisers to reductive subgroups. The results illuminate the structure underlying the Moretó–Rizo conjecture on character correspondences and have implications for permutation-group questions about quasi-semiregular elements.

Abstract

We determine subnormalisers of semisimple elements of prime power order in finite quasi-simple groups of Lie type. For this, we determine the maximal overgroups of normalisers of Sylow tori. This is motivated by the recent character correspondence conjecture by Moretó and Rizo as well as by the question of existence of quasi-semiregular elements in finite permutation groups.

Subnormalisers of semisimple elements in finite groups of Lie type

TL;DR

This work provides a comprehensive classification of subnormalisers of semisimple p-elements in finite quasi-simple groups of Lie type. It reduces the problem to understanding maximal overgroups of normalisers of Sylow d-tori, and then describes the subnormalisers explicitly in both exceptional and classical groups, using Aschbacher’s framework and Levi-subgroup decompositions. The authors extend the analysis to symmetric and sporadic groups to yield a complete landscape, with a notable algebraic-group construction linking finite subnormalisers to reductive subgroups. The results illuminate the structure underlying the Moretó–Rizo conjecture on character correspondences and have implications for permutation-group questions about quasi-semiregular elements.

Abstract

We determine subnormalisers of semisimple elements of prime power order in finite quasi-simple groups of Lie type. For this, we determine the maximal overgroups of normalisers of Sylow tori. This is motivated by the recent character correspondence conjecture by Moretó and Rizo as well as by the question of existence of quasi-semiregular elements in finite permutation groups.

Paper Structure

This paper contains 11 sections, 32 theorems, 38 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Subnormalisers of semisimple elements in the abelian Sylow case
  3. Overgroups of Sylow tori normalisers and subnormalisers in groups of exceptional type
  4. Overgroups of Sylow tori normalisers and subnormalisers in groups of classical type
  5. The special linear groups
  6. The special unitary groups
  7. The symplectic and odd-dimensional orthogonal groups
  8. The even-dimensional orthogonal groups
  9. An algebraic group construction
  10. Subnormalisers in symmetric groups
  11. On subnormalisers in sporadic groups

Key Result

Theorem 2

Let $G$ be a finite quasi-simple group and $p$ a prime such that $G$ has abelian Sylow $p$-subgroups. Then the subnormalisers of all $p$-elements in $G$ are known.

Theorems & Definitions (71)

  • Conjecture 1: Moretó--Rizo
  • Theorem 2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 61 more