ON THE NEAR FRATTINI SUBGROUPS OF CERTAIN GROUPS

Let lambda(G) and psi(G) represent the lower near Frattini and the near Frattini subgroups of a group G. We prove that if G = A(*H) B is the amalgamated free product of the groups A and B where H is any cyclic subgroup, then lambda(G) less-than-or-equal-to H. We show that psi(G) = 1, if G is: (1) the fundamental group of a two-dimensional orientable surface; (2) an infinite cyclic extension of an n generator (n greater-than-or-equal-to 2) free group; (3) any fibered knot group; (4) any torus knot group. Also, we show that lambda(G) = 1 if. (1) G = A(*H) B, where H is infinite cyclic and is a retract of A and B; (2) G is the knot group of any product of knots; (3) G is a cable knot group. These results are analogous to results on the Frattini subgroups of such groups obtained by R. B. J. T. Allenby and C. Y. Tang, [1], R. B. J. T. Allenby, C. Y. Tang and S. Y. Tang [2], and A. Whittemore [13].