# On the normalizer of an iterated wreath product

## TL;DR

This paper characterizes the normalizer of an iterated wreath product of a group G within the symmetric group on G^n, revealing it as a semi-direct product involving automorphisms of G.

## Contribution

It provides a precise description of the normalizer of iterated wreath products in symmetric groups, including a recursive description of automorphism actions.

## Key findings

- Normalizer equals a semi-direct product involving automorphisms
- Explicit recursive description of automorphism action
- Applicable to both finite and infinite groups

## Abstract

Given a group $G$ and $n\geq 0$, let $W(G,n)$ be the associated iterated wreath product -- unrestricted when $G$ is infinite -- viewed as a permutation group on $G^n$. We prove that the normalizer of $W(G,n)$ in the symmetric group $S(G^n)$ is equal to $M_n\ltimes W(G,n)$, where $M_n$ is isomorphic to~$\mathrm{Aut}(G)^n$. The action of $\mathrm{Aut}(G)^n$ on $W(G,n)$ is recursively described.