On the multiple holomorph of a finite almost simple group

Abstract

Let $G$ be a group. Let $\mathrm{Perm}(G)$ denote its symmetric group and write $\mathrm{Hol}(G)$ for the normalizer of the subgroup of left translations in $\mathrm{Perm}(G)$. The multiple holomorph $\mathrm{NHol}(G)$ of $G$ is in turn defined to be the normalizer of $\mathrm{Hol}(G)$ in $\mathrm{Perm}(G)$. In this paper, we shall show that the quotient group $\mathrm{NHol}(G)/\mathrm{Hol}(G)$ has order two when $G$ is finite and almost simple. As an application of our techniques, we shall also develop a method to count the number of Hopf-Galois structures of isomorphic type on a finite almost simple extension in terms of fixed point free endomorphisms.

arXiv commentrevised based on referee's comments; more details are added to the proof of Theorem 1.3

References in corpus (5)

• Skew braces and the Yang-Baxter equation
• The multiple holomorph of a finitely generated abelian group
• Groups that have the same holomorph as a finite perfect group
• Non-existence of Hopf-Galois structures and bijective crossed homomorphisms
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Cited by in corpus (9)

• Hopf-Galois structures on finite extensions with almost simple Galois group
• On the multiple holomorph of groups of squarefree or odd prime power order
• The multiple holomorph of a semidirect product of groups having coprime exponents
• Hopf-Galois structures on finite extensions with quasisimple Galois group
• Finite $p$-groups of class two with a large multiple holomorph
• The multiple holomorph of split metacyclic $p$-groups
• Normalizer Quotients of Symmetric Groups and Inner Holomorphs
• Non-abelian simple groups which occur as the type of a Hopf-Galois structure on a solvable extension
• The NNN-Property of Cyclic Groups