End OrbitStabilizer.

(* Specifications: *)
Lemma card_orbit_in G x : G \subset D -> #|orbit to G x| = #|G : 'C_G[x | to]|.
Lemma card_orbit_in_stab G x :
G \subset D -> (#|orbit to G x| * #|'C_G[x | to]|)%N = #|G|.
Lemma astab1_act_in x a : a \in D -> 'C[to x a | to] = 'C[x | to] :^ a.
Lemma atrans_dvd_index_in G S :
G \subset D -> [transitive G, on S | to] -> #|S| %| #|G : 'C_G(S | to)|.
Lemma atrans_dvd_in G S :
G \subset D -> [transitive G, on S | to] -> #|S| %| #|G|.
Lemma atransPin G S :
G \subset D -> [transitive G, on S | to] ->
forall x, x \in S -> orbit to G x = S.
Lemma subgroup_transitivePin G H S x :
x \in S -> H \subset G -> G \subset D -> [transitive G, on S | to] ->
reflect ('C_G[x | to] * H = G) [transitive H, on S | to].