13.5 (1995)

Groups $A$ and $B$ are said to be locally equivalent if for every finitely generated subgroup $X \leqslant A$ there is a subgroup $Y \leqslant B$ isomorphic to $X$, and conversely, for every finitely generated subgroup $Y \leqslant B$ there is a subgroup $X \leqslant A$, isomorphic to $Y$. We call a group $G$ categorical if $G$ is isomorphic to any group that is locally equivalent to $G$. Is it true that a periodic locally soluble group $G$ is categorical if and only if $G$ is hyperfinite with Chernikov Sylow subgroups? (A group is hyperfinite if it has a well ordered ascending normal series with finite factors.)