The multiple holomorph of split metacyclic $p$-groups

Abstract

Given any group $G$, the normalizer $\mathrm{Hol}(G)$ of the subgroup of left translations in the group of all permutations on $G$ is called the holomorph, and the normalizer $\mathrm{NHol}(G)$ of $\mathrm{Hol}(G)$ in turn is called the multiple holomorph. The quotient $T(G)=\mathrm{NHol}(G)/\mathrm{Hol}(G)$ has been computed for various families of groups $G$ in the literature. In this paper, we shall supplement the existing results by considering finite split metacyclic $p$-groups $G$ with $p$ an odd prime. We are able to give a closed formula for the order of $T(G)$ when $G$ satisfies some mild conditions. Our work gives a new family of groups $G$ for which $T(G)$ is not a $2$-group.

very minor changes

References in corpus (6)

• The multiple holomorph of a finitely generated abelian group
• Groups that have the same holomorph as a finite perfect group
• The Multiple Holomorphs of Finite $p$-Groups of Class Two
• On the multiple holomorph of a finite almost simple group
• On the multiple holomorph of groups of squarefree or odd prime power order
• The multiple holomorph of a semidirect product of groups having coprime exponents

Cited by in corpus (2)

• Finite $p$-groups of class two with a large multiple holomorph
• The multiple holomorph of centerless groups