The Schur property and groups with uniform conjugacy classes
A well known theorem of Schur states that if \(G/Z(G)\) is finite then so is \(G'\). The authors say that a class of groups \(\mathfrak X\) is a Schur class if whenever \(G/Z(G) \in {\mathfrak X}\) it follows that \(G' \in {\mathfrak X}\). It is also known that polycyclic groups, Chernikov groups and soluble minimax groups are Schur classes. The first part of this paper considers classes of groups with rank restrictions showing in particular that soluble-by-finite groups of finite rank and soluble-by-finite groups with finite section rank are both Schur classes. The second part of the paper is concerned with generalizing B. H. Neumann's result that a group with boundedly finite conjugacy classes has \(G'\) finite. A group is said to have uniform \(\mathfrak X\)-conjugacy classes if there is an \(\mathfrak X\)-group \(Q\) such that \(G/C_G(x^G)\) is isomorphic to a section of \(Q\) for each \(x \in G\). If a class \(\mathfrak X\) is such that any group with uniform \(\mathfrak X\)-conjugacy classes has \(G' \in {\mathfrak X}\), then \(\mathfrak X\) must be a Schur class. However not all Schur classes satisfy this property, there being counterexamples for \(\mathfrak X\) either polycyclic or minimax. Here the authors prove the nontrivial result that the class of groups with finite section rank does have this property and, as a corollary, so does the class of Chernikov groups.
scientific article; zbMATH DE number 783892