On the type-definability of the binding group in simple theories

Abstract Let T be simple, work in C eq over a boundedly closed set. Let p Є S (∅) be internal in a quasi-stably-embedded type-definable set Q (e.g., Q is definable or stably-embedded) and suppose ( p, Q ) is ACL-embedded in Q (see definitions below). Then Aut( p/Q ) with its action on p c is type-definable in C eq over ∅. In particular, if p Є S (∅) is internal in a stably-embedded type-definable set Q , and p c ⋃ Q is stably-embedded, then Aut( p/Q ) is type-definable with its action on p c .