On the Frattini subgroup of a residually finite generalized free product

Abstract:

Let $G = {(A \ast B)_H}$ be the generalized free product of the groups $A,B$ amalgamating the subgroup $H$, and let $\Phi (G)$ denote its Frattini subgroup. In support of the conjecture that $\Phi (G) \subseteq H$ whenever $G$ is resiually finite and $H$ satisfies a nontrivial identical relation, we show, amongst several other things, that the above inequality is indeed valid if in addition at least one of the following holds: (i) $A,B$, each satisfies a nontrivial identical relation; (ii) $G$ is finitely generated; (iii) $H$ is nilpotent. In particular (i) completes earlier investigations of the second author. The methods of proof are, however, different.