Theorem on groups of finite special rank (Q1813271)

A group \(G\) is called locally graduated if every non-trivial finitely generated subgroup of \(G\) has a non-trivial subgroup of finite index. Let \(\mathfrak X\) be the minimal class of groups containing the class of all periodic locally graduated groups and closed under the formation of local systems, sub-Cartesian products, and ascending and descending normal series. It is shown that any group in \(\mathfrak X\) of finite rank (in the sense of Prüfer) has a hyperabelian and a locally soluble subgroup of finite index. This extends several known results.