Conjectures and Questions Regarding Near Frattini Subgroups of 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), μ(G), ψ(G), K(G,H), and Tor H represent the lower near Frattini subgroup of G, the upper near Frattini subgroup of G, the near Frattini subgroup of G, the core of H in G, and the torsion subgroup of H, respectively. Since 1990 a series of papers have been published by the author dealing with the location of λ(G) and ψ(G). We state the main results from these papers for the cases where ψ(G) = 1, ψ(G) ≤ H, ψ(G) ≤ K(G,H), ψ(G) = Tor H, λ(G) = 1, λ(G) ≤ H, λ(G) ≤ K(G,H), or λ(G) = K(G,H). Also, we state the results obtained by Allenby since 1999 for the cases where μ(G) ≤ H, ψ(G) = K(G,H), or ψ(G) = H. The main goal of this paper is to state four conjectures and to pose sixteen related questions for the reader. Mathematics Subject Classification: Primary 20E06, 20E28; Secondary 22, 55