Supersimple $\omega$-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-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.