Some remarks on generalized Schur pairs (Q1265455)

Let \(\Phi\) be a set of words in a free group, \(\mathcal X\) a class of groups and \(G\) a group with a normal subgroup \(K\). Let \(\Phi(K,G)\) be the subgroup generated by the set \[ \{\phi(g_1,\dots,g_{i-1},g_ik,g_{i+1},\dots,g_n)\phi(g_1,\dots,g_n)^{-1};\;\phi\in\Phi,\;g_i\in G,\;k\in K,\;1\leq i\leq n\}, \] and let \[ \begin{multlined}\Phi^*(K,G)=\{k\in K;\;\phi(g_1,\dots,g_{i-1},g_ik,g_{i+1},\dots,g_n)=\phi(g_1,\dots,g_n)\\ \forall\phi\in\Phi,\;g_i\in G,\;1\leq i \leq n\}.\end{multlined} \] The pair \((\Phi,{\mathcal X})\) is said to be an ultra weak Schur pair if whenever both \(K/\Phi^*(K,G)\) and \(K/\Phi(K,G)\) belong to \(\mathcal X\), so does \(\Phi(K,G)\). It is a weak Schur pair if this holds in the case where \(K=G\), and a Schur pair if \(K=G\) and only the first inclusion is assumed. Various properties of ultra weak Schur pairs, analogous to those of Schur pairs and weak Schur pairs are proved, and it is shown that if \(\Phi=\{\psi\}\), where \(\psi\) is an outer commutator word, and \(G\) satisfies the maximum condition, then the finiteness of \(\Phi(K,G)\) implies that of \(K/\Phi^*(K,G)\). -- This is a generalization of a result of \textit{R. F. Turner-Smith} [Proc. Lond. Math. Soc., III. Ser. 14, 321-341 (1964; Zbl 0122.26906)], related to a conjecture of \textit{Philip Hall} [J. Reine Angew. Math. 182, 130-141 (1940; Zbl 0023.21001)].