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).
End SymAltDef.
Notation "''Sym_' T" := (Sym (Phant T))
(at level 8, T at level 2, format "''Sym_' T") : group_scope.
Notation "''Sym_' T" := (Sym_group (Phant T)) : Group_scope.
Notation "''Alt_' T" := (Alt (Phant T))
(at level 8, T at level 2, format "''Alt_' T") : group_scope.
Notation "''Alt_' T" := (Alt_group (Phant T)) : Group_scope.

Lemma not_simple_Alt_4 (T : finType) : #|T| = 4 -> ~~ simple 'Alt_T.