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Abelian supplements in almost simple groups

Mauro Costantini, Andrea Lucchini, Daniele Nemmi

TL;DR

The paper addresses the structure of finite almost simple groups $G$ with socle $G_0$ when the quotient $G/G_0$ is abelian, proving the existence of an abelian subgroup $A$ such that $G=AG_0$. The authors reduce the problem to constructing abelian supplements inside the outer automorphism group $ ext{Out}(G_0)$, introducing and exploiting the concept of $T$-abelian supplements, and they develop explicit constructions across all families of simple groups of Lie type (linear, unitary, and exceptional types). The main contributions are case-by-case constructions of abelian supplements for all abelian subgroups $T rianglelefteq ext{Out}(G_0)$, yielding the overarching result and enabling applications such as corollaries on the generalized Fitting subgroup and on generating graphs. These results enhance understanding of how abelian extensions interact with the core simple structure, with implications for group generation and automorphism-induced decompositions in almost simple groups, and they provide concrete tools for analyzing related combinatorial properties. $G=AG_0$ is thus attainable for all such almost simple groups, with the abelian supplement explicitly realized in terms of automorphisms and explicit matrix realizations in the Lie-type settings.

Abstract

Let $G$ be a finite almost simple group with socle $G_0$. In this paper we prove that whenever $G/G_0$ is abelian, then there exists an abelian subgroup $A$ of $G$ such that $G=AG_0$. We propose a few applications of this structural property of almost simple groups.

Abelian supplements in almost simple groups

TL;DR

The paper addresses the structure of finite almost simple groups with socle when the quotient is abelian, proving the existence of an abelian subgroup such that . The authors reduce the problem to constructing abelian supplements inside the outer automorphism group , introducing and exploiting the concept of -abelian supplements, and they develop explicit constructions across all families of simple groups of Lie type (linear, unitary, and exceptional types). The main contributions are case-by-case constructions of abelian supplements for all abelian subgroups , yielding the overarching result and enabling applications such as corollaries on the generalized Fitting subgroup and on generating graphs. These results enhance understanding of how abelian extensions interact with the core simple structure, with implications for group generation and automorphism-induced decompositions in almost simple groups, and they provide concrete tools for analyzing related combinatorial properties. is thus attainable for all such almost simple groups, with the abelian supplement explicitly realized in terms of automorphisms and explicit matrix realizations in the Lie-type settings.

Abstract

Let be a finite almost simple group with socle . In this paper we prove that whenever is abelian, then there exists an abelian subgroup of such that . We propose a few applications of this structural property of almost simple groups.
Paper Structure (16 sections, 25 theorems, 253 equations, 2 tables)

This paper contains 16 sections, 25 theorems, 253 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminary results
  3. Linear groups
  4. Unitary groups
  5. Notation for groups of Lie type
  6. $C_n(q)$, $B_n(q)$, $n\geqslant 2$, $E_7(q)$
  7. $E_6(q)$
  8. $p\equiv 1\mod 3$, $T=\langle \delta,\varphi\rangle$
  9. $p\equiv -1\mod 3$, $T=\langle \delta, \varphi\tau\rangle$
  10. ${}^2\!E_6(q)$
  11. ${}^2\!D_n(q)$, $n$ even
  12. $D_n(q)$, $n\geqslant 3$, $n$ odd
  13. $p\equiv1 \mod{4}$
  14. $p\equiv-1 \mod{4}$
  15. $D_n(q)$, $n$ even
  16. ...and 1 more sections

Key Result

Proposition 1

Let $\mathcal{C}$ be a class of finite groups which satisfies the following properties: If $N$ is a normal subgroup of a finite group $G$ and $G/N\in \mathcal{C},$ then there exists $H\in \mathcal{C}$ such that $G=HN.$

Theorems & Definitions (41)

  • Proposition 1
  • proof
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 31 more