intuition.
rewrite <- (group_cyclic g x).
rewrite <- (@groupIdent _ groupOp ident _ G _ _ _ FCG g).
eapply groupExp_eq_if; trivial.
apply g_generator.
rewrite <- (group_cyclic g y) in H; eauto.
rewrite <- (group_cyclic g x) in H; eauto.
rewrite <- groupExp_plus in H.
apply groupExp_eq in H; trivial.
eapply modNat_add_same_r.
eauto.
apply g_generator.
apply g_generator.
apply g_generator.
apply g_generator.
apply g_generator.
Qed.

Theorem groupExp_mod : forall g n,
generator g ->
g^n = g^(modNat n order).