Definition Z_to_group : Hom (ZZ:GROUP) G.Definition group_power (G : GROUP) (x : G) (n : ZZ) := Z_to_group x n.Definition sgroup_powers (G : GROUP) (g : G) := coKer (Z_to_group g).

(* Specifications: *)
Lemma sgroup_powers_prop : forall (G : GROUP) (g x : G), in_part x (sgroup_powers g) -> exists n : ZZ, Equal x (group_power G g n).
Lemma sgroup_powers_rev : forall (G : GROUP) (g : G) (n : ZZ), in_part (group_power G g n) (sgroup_powers g).