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(*  Reflected boolean property and morphic image, injection, bijection *)
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Section MorphicImage.
Variables aT rT : finGroupType.
Variables (D : {group aT}) (f : {morphism D >-> rT}) (x : aT).
Hypothesis Dx : x \in D.

(* Specifications: *)
Lemma morph_order : #[f x] %| #[x].
Lemma cyclicJ G x : cyclic (G :^ x) = cyclic G.
Lemma eq_subG_cyclic G H K :
cyclic G -> H \subset G -> K \subset G -> (H :==: K) = (#|H| == #|K|).
Lemma cardSg_cyclic G H K :
cyclic G -> H \subset G -> K \subset G -> (#|H| %| #|K|) = (H \subset K).
Lemma sub_cyclic_char G H : cyclic G -> (H \char G) = (H \subset G).
Lemma morphim_cyclic rT G H (f : {morphism G >-> rT}) :
cyclic H -> cyclic (f @* H).
Lemma quotient_cycle x H : x \in 'N(H) -> <[x]> / H = <[coset H x]>.
Lemma dvdn_prime_cyclic G p : prime p -> #|G| %| p -> cyclic G.
Lemma cyclic_small G : #|G| <= 3 -> cyclic G.
End CyclicProps.