Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Elementary abelian subgroups: from algebraic groups to finite groups

Jianbei An, Heiko Dietrich, Alastair J. Litterick

Abstract

We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian $p$-subgroups for torsion primes $p$ in finite groups of exceptional Lie type. Such classification results are important for determining the maximal $p$-local subgroups and $p$-radical subgroups, both of which play a crucial role in modular representation theory.

Elementary abelian subgroups: from algebraic groups to finite groups

Abstract

We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian -subgroups for torsion primes in finite groups of exceptional Lie type. Such classification results are important for determining the maximal -local subgroups and -radical subgroups, both of which play a crucial role in modular representation theory.
Paper Structure (31 sections, 21 theorems, 40 equations, 1 figure, 10 tables)

This paper contains 31 sections, 21 theorems, 40 equations, 1 figure, 10 tables.

Table of Contents

  1. Introduction
  2. Main results and structure of the paper
  3. Notation
  4. Preliminaries
  5. Independence from the characteristic
  6. Torality
  7. Semisimple elements of small order
  8. From algebraic groups to finite groups
  9. The correspondence
  10. Local structure
  11. The $F$-action on $N_G(A)/C_G(A)^\circ$
  12. Cohomology
  13. Toral subgroups of simple algebraic groups
  14. Computation of toral elementary abelian subgroups
  15. Translation to finite groups
  16. ...and 16 more sections

Key Result

Proposition 3.1

Let $\ell$ be a prime number and $G$ a semisimple split group scheme; let $X$ be a finite group of order coprime to $\ell$. For any algebraically closed fields $K_0$ and $K_\ell$ of characteristic $0$ and $\ell$, respectively, the sets $\mathop{\mathrm{Hom}}\nolimits(X,G(K_{0}))/G(K_{0})$ and $\math

Figures (1)

  • Figure 1: Hasse diagram of all non-toral $2$-subgroups of $E_{7,{{\text{\rm ad}}}}$; cross-family inclusions in grey.

Theorems & Definitions (40)

  • Remark 1.1
  • Proposition 3.1: GrRy
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4: SteinbergTorsion
  • Lemma 3.5
  • proof
  • Proposition 4.1
  • ...and 30 more