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Finite almost simple groups whose holomorph contains a solvable regular subgroup

Cindy Tsang

Abstract

In our previous paper, we gave a complete list of the finite non-abelian simple groups whose holomorph contains a solvable regular subgroup. In this paper, we refine our previous work by considering all finite almost simple groups. In particular, our result yields a complete characterization of the finite almost simple groups which occur as the type of a Hopf-Galois structure on a solvable extension, or equivalently, the additive group of a skew brace having a solvable multiplicative group.

Finite almost simple groups whose holomorph contains a solvable regular subgroup

Abstract

In our previous paper, we gave a complete list of the finite non-abelian simple groups whose holomorph contains a solvable regular subgroup. In this paper, we refine our previous work by considering all finite almost simple groups. In particular, our result yields a complete characterization of the finite almost simple groups which occur as the type of a Hopf-Galois structure on a solvable extension, or equivalently, the additive group of a skew brace having a solvable multiplicative group.
Paper Structure (3 sections, 3 theorems, 18 equations)

This paper contains 3 sections, 3 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Proof of cases (2) to (6)
  3. Proof of case (1)

Key Result

Theorem 1.1

Let $N$ be a finite almost simple group. If $(G,N)$ is realizable for some solvable group $G$, then the socle of $N$ is isomorphic to The converse is also true when $N$ is a finite non-abelian simple group.

Theorems & Definitions (4)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 1.4