Structure of finite(ly generated) abelian groups

• AddCommGroup.equiv_free_prod_directSum_zmod : Any finitely generated abelian group is the product of a power of ℤ and a direct sum of some ZMod (p i ^ e i) for some prime powers p i ^ e i.
• AddCommGroup.equiv_directSum_zmod_of_fintype : Any finite abelian group is a direct sum of some ZMod (p i ^ e i) for some prime powers p i ^ e i.

theorem Module.finite_of_fg_torsion (M : Type u) [AddCommGroup M] [Module ℤ M] [Module.Finite ℤ M] (hM : Module.IsTorsion ℤ M) : Finite M

theorem AddCommGroup.equiv_free_prod_directSum_zmod (G : Type u) [AddCommGroup G] [hG : AddGroup.FG G] : ∃ n ι x p x e, Nonempty (G ≃+ (Fin n →₀ ℤ) × ⨁ (i : ι), ZMod (p i ^ e i))

Structure theorem of finitely generated abelian groups : Any finitely generated abelian group is the product of a power of ℤ and a direct sum of some ZMod (p i ^ e i) for some prime powers p i ^ e i.

theorem AddCommGroup.equiv_directSum_zmod_of_fintype (G : Type u) [AddCommGroup G] [Finite G] : ∃ ι x p x e, Nonempty (G ≃+ ⨁ (i : ι), ZMod (p i ^ e i))

Structure theorem of finite abelian groups : Any finite abelian group is a direct sum of some ZMod (p i ^ e i) for some prime powers p i ^ e i.

theorem AddCommGroup.finite_of_fg_torsion (G : Type u) [AddCommGroup G] [hG' : AddGroup.FG G] (hG : AddMonoid.IsTorsion G) : Finite G

theorem CommGroup.finite_of_fg_torsion (G : Type u) [CommGroup G] [Group.FG G] (hG : Monoid.IsTorsion G) : Finite G