8.50 (1982)

At a conference in Oberwolfach in 1979 I exhibited a finitely generated soluble group $G$ (of derived length 3) and a non-zero cyclic $\mathbb{Z}G$-module $V$ such that $V \cong V \oplus V$. Can this be done with $G$ metabelian? I conjecture that it cannot. For background of this problem and for details of the construction see (P. M. Neumann, in: Groups–Korea, Pusan, 1988 (Lect. Notes Math., 1398), Springer, Berlin, 1989, 124–139).