Section SymAltDef.
Variable T : finType.
Implicit Types (s : {perm T}) (x y z : T).
(** Definitions of the alternate groups and some Properties **)
Definition Sym of phant T : {set {perm T}} := setT.
Canonical Sym_group phT := Eval hnf in [group of Sym phT].
Local Notation "'Sym_T" := (Sym (Phant T)) (at level 0).
Canonical sign_morph := @Morphism _ _ 'Sym_T _ (in2W (@odd_permM _)).
Definition Alt of phant T := 'ker (@odd_perm T).
Canonical Alt_group phT := Eval hnf in [group of Alt phT].
Local Notation "'Alt_T" := (Alt (Phant T)) (at level 0).

Lemma card_Alt : 1 < n -> (2 * #|'Alt_T|)%N = n`!.