A key theorem on the near Frattini subgroups of generalized free products of groups

Let G = A star(H)B be the generalized free product of the groups A and B with the amalgamated subgroup H, and let lambda(G) be the lower near Frattini subgroup of G. As the main theorem, we prove that if there exists an element c in G\G such that c centralizes H boolean AND H-c, then lambda(G) less than or equal to H. We use the main theorem to show that if there exists an element c in G\G such that H boolean AND H-c = 1, then lambda(G) = 1. Also, we show that lambda(G) less than or equal to H, provided: (1) there exists a nontrivial normal subgroup N of G such that N boolean AND H = 1; (2) A and B contain finite normal subgroups A(1) and B-1 respectively, such that A(1) boolean AND H = B-1 boolean AND H, and at least one of A(1) or B-1 is not contained in H; (3) H satisfies the maximum condition on subgroups and at least one of A or B is uncountable; (4) A and B are countable groups; (5) H satisfies the minimum or the maximum condition on subgroups. These results are analogous to results bn the Frattini subgroups of such generalized free products of groups achieved by R.B.J.T. Allenby and C.Y. Tang.