Residual finiteness of the free product of two groups with commuting subgroups (Q1307171)

A sufficient condition for residual finiteness of the free product of two groups with amalgamated subgroup is well known [\textit{G.~Baumslag}, Trans. Am. Math. Soc. 106, 193-209 (1963; Zbl 0112.25904)]. However this condition is not necessary. Let \(A\) and \(B\) be groups and let \(H\) and \(K\) be subgroups of \(A\) and \(B\) respectively. The group \(G=(A*B;\;[H,K]=1)\) with generating set the union of the generators of \(A\) and \(B\) and the set of the defining relations of \(A\) and \(B\) and all relations in the set \(\{[h,k]=1\mid h\in H,\;k\in K\}\) is called the free product of \(A\) and \(B\) with commuting subgroups \(H\) and \(K\). The author gives a necessary and sufficient condition for the free product of residually finite groups with commuting subgroups to be residually finite (Theorem 1). In Theorem 2 the author proves that if \(A\) and \(B\) are residually finite \(p\)-groups and \(H\) and \(K\) are nonidentity subgroups of \(A\) and \(B\) respectively, then the group \(G=(A*B;\;[H,K]=1)\) is a residually finite \(p\)-group if and only if the subgroups \(H\) and \(K\) are separated in \(A\) and \(B\) with respect to the class of finite \(p\)-groups. Recall that a subgroup \(X\) of a group \(Y\) is separated with respect to the class of finite \(p\)-groups if, for every element \(y\in Y\setminus X\), there exists a homomorphism \(\rho\) from \(Y\) into some finite \(p\)-group such that \(\rho(y)\notin\rho(X)\).