Mutually Normalizing Regular Permutation Groups and Zappa-Szep Extensions of the Holomorph

Abstract

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