A note on stable sets, groups, and theories with NIP

Let M be an arbitrary structure. We say that an M -formula φ(x) defines a stable set in M if every formula φ(x)∧α(x, y) is stable. We prove: If G is an M -definable group and every definable stable subset of G has U-rank at most n (the same n for all sets) then G has a maximal connected stable normal subgroup H such that G/H is purely unstable. The assumptions holds for example when the structure M is interpretable in an o-minimal structure. More generally, an M -definable set X is called weakly stable if the M -induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP, is stable.