© Hindawi Publishing Corp. NEAR FRATTINI SUBGROUPS OF RESIDUALLY FINITE GENERALIZED FREE PRODUCTS OF GROUPS

Abstract. Let G = AHB be the generalized free product of the groups A and B with the amalgamated subgroup H. Also, let λ(G) and ψ(G) represent the lower near Frattini subgroup and the near Frattini subgroup of G, respectively. If G is finitely generated and residually finite, then we show that ψ(G) ≤ H, provided H satisfies a nontrivial identical relation. Also, we prove that if G is residually finite, then λ(G)≤H, provided: (i)H satisfies a nontrivial identical relation and A,B possess proper subgroups A1,B1 of finite index containing H; (ii) neither A nor B lies in the variety generated by H; (iii) H < A1 ≤ A and H < B1 ≤ B, whereA1 and B1 each satisfies a nontrivial identical relation; (iv)H is nilpotent. 2000 Mathematics Subject Classification. 20E06, 20E28. 1. Definitions and notation. Throughout the paper our notation will be standard. We use G = AHB to represent the generalized free product of the groups A and B with the amalgamated subgroup H, as in B. H. Neumann [12]. A group G is residually finite if every nontrivial element of G can be excluded from some normal subgroup of finite index in G. An N-group is a finite group in which the normalizer of every nontrivial solvable subgroup is solvable. A group G is called 3-metabelian if ever