End InternalActionDefs.
Notation "'R" := (@mulgr_action _) (at level 8) : action_scope.
Notation "'Rs" := (@rcoset_action _) (at level 8) : action_scope.
Notation "'J" := (@conjg_action _) (at level 8) : action_scope.
Notation "'J" := (@conjg_groupAction _) (at level 8) : groupAction_scope.
Notation "'Js" := (@conjsg_action _) (at level 8) : action_scope.
Notation "'JG" := (@conjG_action _) (at level 8) : action_scope.
Notation "'Q" := ('J / _)%act (at level 8) : action_scope.
Notation "'Q" := ('J / _)%gact (at level 8) : groupAction_scope.
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).