Section InternalGroupAction.
Variable gT : finGroupType.
Implicit Types A B : {set gT}.
Implicit Types G H : {group gT}.
Implicit Type x : gT.
(*  Various identities for actions on groups. *)

Lemma sub_astabQR A B H :
A \subset 'N(H) -> B \subset 'N(H) ->
(A \subset 'C(B / H | 'Q)) = ([~: A, B] \subset H).