The holomorph of an extra-special p–group

Abstract

Let G be a group and S(G) be a group of permutations on the set G. Define the holomorph of G to be the normalizer of the image in S(G) of the right regular representation, Hol(G) = NS(G)(ρ(G)) = Aut(G)ρ(G). If N is a regular subgroup of S(G), then NS(G)(N) is isomorphic to the holomorph of G. Miller has shown that the group T (G) = NS(G)(Hol(G))/Hol(G) acts regularly on the set of the regular subgroups N of S(G) which are isomorphic to G, and have the same holomorph as G, in the sense that NS(G)(N) = Hol(G). In this paper we study T (G), when G is an extra-special p-group.