  It is known that a group shift on a polycyclic group is necessarily of finite
type. We show that, for trivial reasons, if a group does not satisfy the
maximal condition on subgroups, then it admits non-SFT abelian group shifts. In
particular, we show that if group is elementarily amenable or satisfies the
Tits alternative, then it is virtually polycyclic if and only if all its group
shifts are of finite type. Our theorems are minor elaborations of results of
Schmidt and Osin.
