The multiple holomorph of centerless groups

Abstract

Let G be a group. The holomorph Hol(G) may be defined as the normalizer of the subgroup of either left or right translations in the group of all permutations of G. The multiple holomorph NHol(G) is in turn defined as the normalizer of the holomorph. Their quotient T (G) = NHol(G)/Hol(G) has been computed for various families of groups G. In this paper, we consider the case when G is centerless, and we shall show that T (G) must be elementary 2-abelian unless G satisfies some fairly strong conditions. As an application of our main theorem, we are able to show that T (G) has order 2 when G is an almost simple group, and T (G) is elementary 2-abelian when G is a centerless perfect or complete group. Contents 1. Introduction 1 2. Preliminaries 4 3. Our main theorem 6 4. Applications 13 References 16 Appendix: Magma code 18