On the residual finiteness of generalised free products of nilpotent groups

Abstract

GILBERT BAUMSLAG(i) 1. Introduction.1.1.Notation.Let c be a class of groups.Then tf denotes the class of those groups which are residually in eS(2), i.e., GeRt if, and only if, for each x e G (x # 1) there is an epimorph of G in ^ such that the element corresponding to x is not the identity.If 2 is another class of groups, then we denote by %> Qt the class of those groups G which possess a normal subgroup N in # such that G/NeSi(f).For convenience we call a group G a Schreier product if G is a generalised free product with one amalgamated subgroup. Let us denote by , . _> o(A,B)the class of all Schreier products of A and B. It is useful to single out certain subclasses of g (A, B) by specifying that the amalgamated subgroup satisfies some condition T, say; we denote this subclass by (A,B; V).Thus o(A,B; V) consists of all those Schreier products of A and B in which the amalgamated subgroup satisfies the condition T.We shall use the letters J5", Jf, and to denote, respectively, the class of finite groups, the class of finitely generated nilpotent groups without elements of finite order, and the class of free groups.1.2.A negative theorem.The simplest residually finite(4) groups are the finitely generated nilpotent groups (K. A. Hirsch [2]).In particular, then, (1.21) Jf c k3F.Since the free product of residually finite groups is residually finite (K.W. Gruenberg [3]), the free product of any given pair of finitely generated nilpotent groups is residually finite.Hence, if A,B eJf,(1.22) o(A,B; trivial) c R&.