Section Injm.
Variables (rT : finGroupType) (D : {group gT}) (f : {morphism D >-> rT}).
Hypothesis injf : 'injm f.

(* Specifications: *)
Lemma injm_center G : G \subset D -> f @* 'Z(G) = 'Z(f @* G).
Lemma center_cprod A B G : A \* B = G -> 'Z(A) \* 'Z(B) = 'Z(G).
Lemma center_bigcprod I r P (F : I -> {set gT}) G :
\big[cprod/1]_(i <- r | P i) F i = G ->
\big[cprod/1]_(i <- r | P i) 'Z(F i) = 'Z(G).
Lemma cprod_center_id G : G \* 'Z(G) = G.
Lemma center_bigdprod I r P (F: I -> {set gT}) G :
\big[dprod/1]_(i <- r | P i) F i = G ->
\big[dprod/1]_(i <- r | P i) 'Z(F i) = 'Z(G).
Lemma Aut_cprod_full G H K :
H \* K = G -> 'Z(H) = 'Z(K) ->
Aut_in (Aut H) 'Z(H) \isog Aut 'Z(H) ->
Aut_in (Aut K) 'Z(K) \isog Aut 'Z(K) ->
Aut_in (Aut G) 'Z(G) \isog Aut 'Z(G).