Near Frattini Subgroups of Certain Generalized Free Products of Groups

Let G = A ∗H B 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 of G and the near Frattini subgroup of G respectively. We show that G is ψ−free provided: (i) G is any ordinary free product of groups; (ii) G = A ∗H B and there exists an element c in G\H such thatH ∩H = 1; (iii) G = A ∗H B and λ(G) ∩H = μ(G) ∩H = 1; (iv) G = A∗HB, where A and B are finitely generated and λ−free, and H = C(∞); (v) G = A ∗H B, and H 6= 1 is malnormal in at least one of A or B; (vi) G is a surface group; (vii) G is the group of an unknotted circle in R3; (viii) G is a group of F−type with only odd torsion where neither U nor V is a proper power; (ix) G is a non-elementary planar discontinuous group with only odd torsion. Furthermore, we show that if G = A ∗H B, then: (i) λ(G) ≤ H, provided both A and B are nilpotent; (ii) ψ(G) ≤ H, provided both A and B are finitely generated and nilpotent. AMS Subject Classification: 20E06, 20E28