Some properties of ultra Hall and Schur pairs
Let \(\Phi\) be a set of words of a free group, then \(\Phi(G)\) and \(\Phi^*(G)\) denote the verbal and marginal subgroups of \(G\) corresponding to \(\Phi\). If \(\mathcal K\) is a class of groups, then \((\Phi,{\mathcal K})\) is said to be a `Schur pair' (`weak Schur pair') if, for every group \(G\), \(G/\Phi^*(G)\in\mathcal K\) (\(G/\Phi^*(G)\in\mathcal K\) and \(G/\Phi(G)\in\mathcal K\)) implies \(\Phi(G)\in\mathcal K\). Let \(\Phi M(G)\) denote the Baer invariant of \(G\), then the first result of this paper shows that \((\Phi,{\mathcal K})\), where \({\mathcal K}=PQS_n\mathcal K\), is a Schur pair if and only if \(G\in\mathcal K\) implies \(\Phi M(G)\in\mathcal K\). The concept of an ultra Hall pair is introduced and it is shown that, if \({\mathcal K}=NPQS_n\mathcal K\) and \((\gamma_2,{\mathcal K})\) is a Schur pair, then \((\phi,{\mathcal K})\) is an ultra Hall pair, for every outer commutator word \(\phi\). Corollaries of both theorems are given.
scientific article; zbMATH DE number 1742824