On the Frattini subgroups of generalized free products and the embedding of amalgams

In this paper we shall prove a basic relation between the Frattini subgroup of the generalized free product of an amalgam $\mathfrak {A} = (A,B;H)$ and the embedding of $\mathfrak {A}$ into nonisomorphic groups, namely, if $\mathfrak {A}$ can be embedded into two non-isomorphic groups ${G_1} = \langle A,B\rangle$ and ${G_2} = \langle A,B\rangle$ then the Frattini subgroup of $G = {(A \ast B)_H}$ is contained in $H$. We apply this result to various cases. In particular, we show that if $A,B$ are locally solvable and $H$ is infinite cyclic then $\Phi (G)$ is contained in $H$.