Supersimple ω-categorical groups and theories

An omega-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl(empty set)-definable subgroup. Every finitely based regular type in a CM-trivial omega-categorical simple theory is non-orthogonal to a type of SU-rank 1. In particular, a supersimple omega-categorical CM-trivial theory has finite SU-rank.