End Basic_commutator_properties.
(***** Set theoretic commutators *****)
Section Commutator_properties.
Variable gT : finGroupType.
Implicit Type (rT : finGroupType) (A B C : {set gT}) (D G H K : {group gT}).

(* Aschbacher, exercise 3.6 (used in proofs of Aschbacher 24.7 and B & G 1.10 *)
Lemma comm_norm_cent_cent H G K :
H \subset 'N(G) -> H \subset 'C(K) -> G \subset 'N(K) ->
[~: G, H] \subset 'C(K).