Locally (soluble-by-finite) groups with all proper insoluble subgroups of finite rank
A group \(G\) has finite rank \(r\) if every finitely generated subgroup of \(G\) is at most \(r\)-generator. If \(\mathfrak C\) is a class of groups then we let \(\mathfrak C^*\) denote the class of groups \(G\) in which every proper subgroup of \(G\) is either of finite rank or in \(\mathfrak C\). We let \(\mathfrak S\) denote the class of soluble groups and \({\mathfrak S}_d\) the class of soluble groups of derived length at most \(d\), where \(d\) is a positive integer. We let \(\Lambda\) denote the set of closure operations \(\{{\mathbf L},{\mathbf R},\text{\textbfṔ},\text{\textbf{\`P}}\}\) and let \(\mathfrak X\) denote the \(\Lambda\)-closure of the class of periodic locally graded groups. Amongst other results we prove that a soluble \({\mathfrak S}^*_d\)-group is either of finite rank or of derived length at most \(d\) and also that a group in the class \({\mathfrak X}\cap({\mathbf L}{\mathfrak S})^*\) is either locally soluble, or has finite rank, or is isomorphic to one of \(\text{SL}(2,\mathbb{F})\), \(\text{PSL}(2,\mathbb{F})\) or \(Sz(\mathbb{F})\) for suitable locally finite fields \(\mathbb{F}\).
scientific article; zbMATH DE number 1026264