A CRITERION FOR HNN EXTENSIONS OF FINITE p-GROUPS TO BE RESIDUALLY p

We give a criterion for an HNN extension of a finite p-group to be residually p. 1. Statement of the Main Results By an HNN pair we mean a pair (G,φ) where G is a group and φ : A → B is an isomorphism between subgroups A and B of G. Given such an HNN pair (G,φ) we consider the corresponding HNN extension G∗ = 〈G, t | t−1at = φ(a), a ∈ A〉 of G, which we denote, by slight abuse of notation, as G∗ = 〈G, t | t−1At = φ(A)〉. Throughout this paper we fix a prime number p, and by a p-group we mean a finite group of p-power order. We are interested in the question under which conditions an HNN extension of a p-group is residually a p-group. (HNN extensions of finite groups are always residually finite [BT78, Co77].) Recall that given a property P of groups, a group G is said to be residually P if for any non-trivial g ∈ G there exists a morphism α : G→ P to a group P with property P such that α(g) is non-trivial. Given HNN pairs (G,φ) and (G′, φ′), a group morphism α : G → G′ is a morphism of HNN pairs if α(A) ⊆ A′, α(B) ⊆ B′, and the diagram A′ φ′ // B′