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 $\mathrm {Hol}(G) = \rho (G)\rtimes \mathrm {Aut}(G)$ 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 $G = A \rtimes C_d$, where A is a group of finite exponent coprime to d and $C_d$ is the cyclic group of order d.