intros x H'; try assumption.
apply (subgroup_prop (G:=G) (s:=H)); auto with algebra.
Qed.
Definition inj_subgroup : Hom (H:GROUP) G.
apply (Build_monoid_hom (E:=H) (F:=G) (monoid_sgroup_hom:=inj_subsgroup H)).
red in |- *.
auto with algebra.
Defined.

Lemma inj_subgroup_injective : injective inj_subgroup.
red in |- *.