Normalizer quotients of symmetric groups and inner holomorphs

Abstract

We show that every finite group T is isomorphic to a normalizer quotient $N_{S_n}\left(H\right)/H$ for some n and a subgroup $H\le S_n$. We show that this holds for all large enough $n\ge n_0\left(T\right)$ 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 $\mathrm{Out}\left(G\right)\cong T$ (for any given finite group T) and the determination of the normalizer in $\mathrm{Sym}\left(G\right)$ of the inner holomorph $\mathrm{InHol}\left(G\right)\le \mathrm{Sym}\left(G\right)$ for any centerless indecomposable finite group G, which may be of independent interest.