# Documentation

Mathlib.GroupTheory.FiniteAbelian

# 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) [] [] [] (hM : ) :
theorem AddCommGroup.equiv_free_prod_directSum_zmod (G : Type u) [] [hG : ] :
n ι x p x e, Nonempty (G ≃+ (Fin n →₀ ) × ⨁ (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) [] [] :
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) [] [hG' : ] (hG : ) :
theorem CommGroup.finite_of_fg_torsion (G : Type u) [] [] (hG : ) :