Mutually normalizing regular permutation groups and Zappa–Szép extensions of the holomorph

Abstract

For a group $G$, embedded in its group of permutations $B=\text{Perm}(G)$ via the left regular representation $\lambda :G\to B$, the normalizer of $\lambda (G)$ in $B$ is $\text{Hol}(G)$, the holomorph of $G$. The set $\mathcal{\mathscr{H}}(G)$ of those regular $N\le \text{Hol}(G)$ such that $N\cong G$ and ${\text{Norm}}_B(N)=\text{Hol}(G)$ is keyed to the structure of the so-called multiple holomorph of $G$, $\text{NHol}(\text{G})={\text{Norm}}_B(\text{Hol}(G))$, in that $\mathcal{\mathscr{H}}(G)$ is the set of conjugates of $\lambda (G)$ by $\text{NHol}(G)$. We wish to generalize this by considering a certain set $\mathcal{𝒬}(G)$ consisting of regular subgroups $M\le \text{Hol}(G)$, where $M\cong G$, that contains $\mathcal{\mathscr{H}}(G)$ with the property that its members mutually normalize each other. This set will generally give rise to a group $Q\text{Hol}(G)$ which we will call the quasi-holomorph of $G$, where the orbit of $\lambda (G)$ under $Q\text{Hol}(G)$ is $\mathcal{𝒬}(G)$. The multiple holomorph is a group extension of $\text{Hol}(G)$ and the quasi-holomorph will contain $\text{NHol}(\text{G})$, but, when larger than $\text{NHol}(\text{G})$, is frequently a Zappa–Szép product with the holomorph.