End NTransitveProp.
Section NTransitveProp1.
Variables (gT : finGroupType) (sT : finType).
Variables (to : {action gT &-> sT}) (G : {group gT}) (S : {set sT}).

(* Specifications: *)
(* This is the forward implication of Aschbacher (15.12).1  *)
Theorem stab_ntransitive m x :
0 < m -> x \in S -> [transitive^m.+1 G, on S | to] ->
[transitive^m 'C_G[x | to], on S :\ x | to].
(* This is the converse implication of Aschbacher (15.12).1  *)
Theorem stab_ntransitiveI m x :
x \in S -> [transitive G, on S | to] ->
[transitive^m 'C_G[x | to], on S :\ x | to] ->
[transitive^m.+1 G, on S | to].