The multiple holomorph of a semidirect product of groups having coprime exponents

Abstract

Given any group $G$, the multiple holomorph $\mathrm{NHol}(G)$ is the normalizer of the holomorph $\mathrm{Hol}(G)=\rho (G)⋊\mathrm{Aut}(G)$ in the group of all permutations of $G$, where $\rho$ denotes the right regular representation. The quotient $T(G)=\mathrm{NHol}(G)/\mathrm{Hol}G)$ has order a power of $2$ in many of the known cases, but there are exceptions. We shall give a new method of constructing elements (of odd order) in $T(G)$ when $G=A⋊C_d$, where $A$ is a group of finite exponent coprime to $d$ and $C_d$ is the cyclic group of order $d$.

arXiv comment12 pages

References in corpus (5)

• 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

Cited by in corpus (3)

• The multiple holomorph of split metacyclic $p$-groups
• The multiple holomorph of centerless groups
• Hopf--Galois structures of cyclic type on parallel extensions of prime power degree