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