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

Let M be an arbitrary structure. Then 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 hold for example if M is interpretable in an o-minimal structure. More generally, an M-definable set X is 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.