Canonical astabs_group S := group (group_set_astabs S).

(* Specifications: *)
Lemma astab_norm S : 'N(S | to) \subset 'N('C(S | to)).
Lemma astab_normal S : 'C(S | to) <| 'N(S | to).
Lemma acts_orbit G x : G \subset D -> [acts G, on orbit to G x | to].
Lemma acts_subnorm_fix A : [acts 'N_D(A), on 'Fix_to(D :&: A) | to].
Lemma atrans_orbit G x : [transitive G, on orbit to G x | to].
Lemma amove_orbit : amove to G x @: orbit to G x = rcosets 'C_G[x | to] G.
Lemma amoveK :
{in orbit to G x, cancel (amove to G x) (fun Ca => to x (repr Ca))}.
Lemma orbit_stabilizer :
orbit to G x = [set to x (repr Ca) | Ca in rcosets 'C_G[x | to] G].
Lemma act_reprK :
{in rcosets 'C_G[x | to] G, cancel (to x \o repr) (amove to G x)}.