Definition Zgroup (gT : finGroupType) (A : {set gT}) :=
[forall (V : {group gT} | Sylow A V), cyclic V].
Section Zgroups.
Variables (gT rT : finGroupType) (D : {group gT}) (f : {morphism D >-> rT}).
Implicit Types G H K : {group gT}.

Lemma morphim_Zgroup G : Zgroup G -> Zgroup (f @* G).