End MorphGroupAction.
(* Conjugation and right translation actions. *)
Section InternalActionDefs.
Variable gT : finGroupType.
Implicit Type A : {set gT}.
Implicit Type G : {group gT}.
(* This is not a Canonical action because it is seldom used, and it would     *)
(* cause too many spurious matches (any group product would be viewed as an   *)
(* action!).                                                                  *)
Definition mulgr_action := TotalAction (@mulg1 gT) (@mulgA gT).
Canonical conjg_action := TotalAction (@conjg1 gT) (@conjgM gT).

(* Specifications: *)
Lemma conjg_is_groupAction : is_groupAction setT conjg_action.
Canonical conjg_groupAction := GroupAction conjg_is_groupAction.
Lemma rcoset_is_action : is_action setT (@rcoset gT).
Canonical rcoset_action := Action rcoset_is_action.