Normalizer Quotients of Symmetric Groups and Inner Holomorphs

Abstract

We show that every finite group $T$ is isomorphic to a normalizer quotient $N_{S_n}(H)/H$ for some $n$ and a subgroup $H\le S_n$. We show that this holds for all large enough $n\ge n_0(T)$ and also with $S_n$ replaced by $A_n$. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group $G$ with $\mathrm{Out}(G)\cong T$ (for any given finite group $T$) and the determination of the normalizer in $\mathrm{Sym}(\mathrm{G})$ of the inner holomorph $\mathrm{InHol}(G)\le \mathrm{Sym}(G)$ for any centerless indecomposable finite group $G$, which may be of independent interest.

References in corpus (6)

• The multiple holomorph of a finitely generated abelian group
• Groups that have the same holomorph as a finite perfect group
• On the multiple holomorph of a finite almost simple group
• On the multiple holomorph of groups of squarefree or odd prime power order
• Finite $p$-groups of class two with a large multiple holomorph
• Non-abelian simple groups which occur as the type of a Hopf-Galois structure on a solvable extension