17.101 (2010)

According to P. Hall, a group $G$ is said to be homogeneous if every isomorphism of its finitely generated subgroups is induced by an automorphism of $G$. It is known (Higman–Neumann–Neumann) that every group is embeddable into a homogeneous one. Is the same true for the category of representations of groups? A representation $(V, G)$ is finitely generated if $G$ is a finitely generated group, and $V$ a finitely generated module over the group algebra of $G$.