On the frattini subgroups of generalized free products

Abstract

In [6] Higman and N eumann showed that the Frattini subgroup @(A * B) of the ordinary free product of nontrivial groups A and B is trivial and they asked (implicitly) whether or not the Frattini subgroup of a generalized free product G = A *H B of groups A, B with amalgamated subgroup H is always to be found inside H. Whittemore [16] took the first steps toward answering this problem. Subsequently her results have been fairly substantially generalized in a series of papers (e.g., [I, 2, 4, 151). This paper first offers further evidence to support the conjecture that Q(G) < H in all cases and then turns to the problem of the location of the Frattini subgroup of HNN groups. One notable corollary of this latter result is Corollary 6.2. As yet, however, the authors have been unable to see if the results they have obtained are of any value in trying to settle the problem of Neuwirth as to whether the Frattini subgroup of a knot group is trivial1 In Section 2 we first show that the two results which have to date been central in the theory are both corollaries of another rather artifical-looking theorem (Theorem 2.1) but the full hypothesis of the statement of Theorem 2.1 is used in proving, in Section 3, that if G = A *H B and if H has the maximum condition on subgroups then D(G) < H. This result itself depends on the result, found in Section 4 that if A, B are countable and His an arbitrary (countable) group,