Canonical actr_groupAction := GroupAction actr_is_groupAction.
Notation "''M'" := actr_groupAction (at level 8) : groupAction_scope.

(* Specifications: *)
Lemma actr1 u : u ^@ 1 = u.
Lemma actrK x : cancel (actr^~ x) (actr^~ x^-1%g).
Lemma actrKV x : cancel (actr^~ x^-1%g) (actr^~ x).
Theorem Gaschutz_transitive : {in [complements to H in G] &,
forall K L,  K :&: P = L :&: P -> exists2 x, x \in H & L :=: K :^ x}.