Definition Fn_group (F : group) (n : Nat) : GROUP :=
  Build_group
    (Build_group_on (group_inverse_map:=Fn_inv F n)
       (Fn_inv_is_r_inverse (F:=F) (n:=n))
       (Fn_inv_is_l_inverse (F:=F) (n:=n))).

Lemma Fn_group_is_abelian : forall (F : abelian_group) (n : Nat), abelian_group_on (Fn_group F n).