Introduction: In the papers listed in the bibliography and the references therein, Tang and the present author, and others, have shown that both the Frattini subgroup \(\Phi(G)\) and the upper near Frattini subgroup \(\mu(G)\) of a generalized free product \(G=A*_HB\) (where we assume \(A\neq H\neq B\)) are, under certain assumptions, contained in \(H\). Here, as a corollary to our main result, we prove that the same conclusion holds if we make the single assumption that \(H\) be countable.
Since preparing this note, the author has learned that T. Gelander and Y. Glasner have obtained similar results (at least for countable generalized free products and, possibly, for other generalized free products) as a corollary of their work on infinite primitive groups. The methods used here are totally different and also more direct for the case of the theorem at hand.