LAUSR

Abstract Let G be a group, and S ( G ) be the group of permutations on the set G. The (abstract) holomorph of G is the natural semidirect product Aut ( G ) G . We will write Hol ( G ) for the normalizer of the image in S ( G ) of the right regular representation of G, Hol ( G ) = N S ( G ) ( ρ ( G ) ) = Aut ( G ) ρ ( G ) ≅ Aut ( G ) G , and also refer to it as the holomorph of G. More generally, if N is any regular subgroup of S ( G ) , then N S ( G ) ( N ) is isomorphic to the holomorph of N. G.A. Miller has shown that the group T ( G ) = N S ( 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 N S ( G ) ( N ) = Hol ( G ) . If G is non-abelian, inversion on G yields an involution in T ( G ) . Other non-abelian regular subgroups N of S ( G ) having the same holomorph as G yield (other) involutions in T ( G ) . In the cases studied in the literature, T ( G ) turns out to be a finite 2-group, which is often elementary abelian. In this paper we exhibit an example of a finite p-group G ( p ) of class 2, for p > 2 a prime, which is the smallest p-group such that T ( G ( p ) ) is non-abelian, and not a 2-group. Moreover, T ( G ( p ) ) is not generated by involutions when p > 3 . More generally, we develop some aspects of a theory of T ( G ) for G a finite p-group of class 2, for p > 2 . In particular, we show that for such a group G there is an element of order p − 1 in T ( G ) , and exhibit examples where | T ( G ) | = p − 1 , and others where T ( G ) contains a large elementary abelian p-subgroup.