Normalizer Quotients of Symmetric Groups and Inner Holomorphs

TL;DR

This paper proves that every finite group can be realized as a normalizer quotient of symmetric groups or alternating groups, using recent constructions and analysis of holomorph normalizers.

Contribution

It establishes that all finite groups appear as normalizer quotients in symmetric groups for sufficiently large n, extending previous understanding of group embeddings.

Findings

Every finite group T is isomorphic to a normalizer quotient in some symmetric group.

The result holds for all sufficiently large n, including in alternating groups.

Determination of normalizer in Sym(G) of the inner holomorph for certain finite groups.

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 $H\leq S_n$. 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 $\mathrm{Sym(G)}$ of the inner holomorph $\mathrm{InHol}(G)\leq\mathrm{Sym}(G)$ for any centerless indecomposable finite group $G$, which may be of independent interest.