On soluble groups with finite homomorphic images of bounded rank
Denote by \({\mathcal F}(K)\) the set of all finite homomorphic images of a group \(K\). A group is minimax if it has a finite subnormal series each factor of which satisfies either the minimality condition or maximality condition for subgroups. Let a finitely generated group \(G\) be the extension of a nilpotent group by an almost polycyclic group, and let \(H\) be a group of finite rank. The author proves that if \({\mathcal F}(G)\subseteq{\mathcal F}(H)\), then \(G\) is a minimax group. In particular, it is shown that if \(G\) is a finitely generated residually finite group, \(H\) is a soluble minimax group and \({\mathcal F}(G)\subseteq{\mathcal F}(H)\) then \(G\) is minimax.
finitely generated residually finite groups
scientific article; zbMATH DE number 830760