Near Frattini subgroups of residually finite generalized free products of groups. (Q5945779)

The author investigates generalized free products of groups \(A\) and \(B\) with amalgamated subgroup \(H\), \(G=A*_HB\), where \(G\) is finitely generated and residually finite and extends work by R. B. J. T. Allenby and C. Y. Tang concerning the relationship between the lower near Frattini subgroup of \(G\), \(\lambda(G)\), and \(H\).NEWLINENEWLINE After stating results from the author's previous work which show conditions under which \(\lambda(G)\) is a subgroup of \(H\) or a subgroup of the core of \(H\) in \(G\), \(K(G,H)\), a result is proved which shows that if \(H\) satisfies a nontrivial identical relation, then the near Frattini subgroup of \(G\), \(\psi(G)\), is a subgroup of \(H\). Several applications to this theorem are described including such cases as where \(H\) is metabelian or 3-metabelian or where \(H\) is nilpotent.NEWLINENEWLINE Having investigated conditions on the amalgamated subgroup, \(H\), for which \(\psi(G)\leq H\), some results are given for different hypotheses on \(A\), \(B\) and \(H\) for which \(\lambda(G)\leq H\). Using an earlier technique of Allenby and Tang, a theorem is proved which states that if \(G=A*_HB\) is residually finite, \(H\) satisfies a nontrivial identical relation, and \(A\) and \(B\) possess subgroups \(A_1\) and \(B_1\) of finite index containing \(H\), then \(\lambda(G)\leq H\). The paper ends with a conjecture that the hypothesis of this theorem may be weakened to simply, \(H\) satisfies a nontrivial identical relation, thus mirroring the result for the near Frattini subgroup of \(G\), \(\psi(G)\).