ON THE LOWER NEAR FRATTINI SUBGROUPS OF AMALGAMATED FREE-PRODUCTS OF GROUPS

Let G = A*(H)B be the amalgamated free product of the groups A and B with the amalgamated subgroup H. Let psi(G), lambda(G), and K(G, H) represent the near Frattini subgroup of G, the lower near Frattini subgroup of G, and the core of H in G respectively. We show that lambda(G) = 1 if: (1) K(G, H) = 1 and H satisfies the minimum conditions on subgroups; (2) lambda(G) AND H = 1; (3) A and B are free groups, H is finitely generated, and at least one of /A : H/ and /B : H/ is infinite. If H satisfies the minimum conditions on subgroups, then we show that lambda(G) less-than-or-equal-to K(G, H). Also, we show that psi(G) less-than-or-equal-to H, provided G satisfies a nontrivial identical relation. Finally, we prove a lemma concerning nearly splitting groups. These results are analogous to results on the Frattini subgroups of such groups obtained by C. Y. Tang [3], [10], and [11].