11.49 (1990)

B. Hartley (Proc. London Math. Soc., 35, no. 1 (1977), 55–75) constructed an example of a non-countable Artinian $\mathbb{Z}G$-module where $G$ is a metabelian group with the minimum condition for normal subgroups. It follows that there exists a non-countable soluble group (of derived length 3) satisfying Min-$n$. The following question arises in connection with this result and with the study of some classes of soluble groups with the weak minimum condition for normal subgroups. Is an Artinian $\mathbb{Z}G$-module countable if $G$ is a soluble group of finite rank (in particular, a minimax group)?