On the multiple holomorph of groups of squarefree or odd prime power order

Let G be a group and write Perm(G) for its symmetric group. Let us define Hol(G) to be the holomorph of G, regarded as a subgroup of Perm(G), and let NHol(G) denote its normalizer. The quotient T(G) = NHol(G)/Hol(G) has been computed for various families of groups G, and in most of the known cases, it turns out to be elementary 2-abelian, except for two groups of order 16, and some groups of odd prime power order and nilpotency class two. In this paper, we shall show that T(G) is elementary 2-abelian for all finite groups G of squarefree order, and that T(G) is not a 2-group for certain finite p-groups G of nilpotency class at most p - 1. (C) 2019 Elsevier Inc. All rights reserved.