Existing in Mathlib · Algebra

Structure Theorem for Finitely Generated Abelian Groups

Every finitely generated abelian group has the same additive structure as a finite-rank free integer group together with finitely many cyclic groups whose orders are prime powers.

finitely generated abelian groups
free abelian groups
torsion
cyclic prime-power groups
direct sums
additive equivalence

Exact theorem

Exact Mathlib statement

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

The theorem at a glance

Finitely Generated Abelian Groups — free rank and prime-power torsion

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory algebra poster splits a finitely generated abelian group through a two-way equivalence into an integer lattice and a finite direct sum of cyclic prime-power rings. — Finite generation forces an additive equivalence with a finite-rank free integer part and finitely many cyclic prime-power components.

Accessible poster text

Read the headline and panel text

ALGEBRA · ABELIAN GROUP STRUCTURE

STRUCTURE THEOREM FOR FINITELY GENERATED ABELIAN GROUPS

G is a finitely generated abelian group

G ≃+ (Fin n →₀ ℤ) × ⨁ᵢ ZMod(pᵢ^{eᵢ})

for some n, finitely many indices i, prime pᵢ, and exponent eᵢ

FREE PART

A finite-rank integer lattice (Fin n →₀ ℤ)

TORSION PART

A finite direct sum of cyclic prime-power groups ZMod(pᵢ^{eᵢ})

HOW THE CHECKED ROUTE MOVES

1 · Regard G as a finitely generated module over ℤ

2 · Apply the module structure theorem over a principal ideal domain

3 · Separate the free summand from cyclic quotient summands

4 · Replace each integer prime-power quotient by the corresponding ZMod group

5 · Assemble an additive equivalence with G

EXACT SCOPE

Existence of an additive equivalence. No uniqueness, canonical ordering, distinct-prime requirement, or decomposition algorithm.

The detailed visual description covers the diagram and its local mathematical labels.

Statement structure

From hypotheses to conclusion

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

One woven abelian group lattice passes through an equivalence gate and separates into straight integer-grid directions and a finite constellation of cyclic prime-power rings. — Every finitely generated abelian group is additively equivalent to a finite free integer part times finitely many cyclic prime-power parts.

Why it matters

A mathematical landmark

The structure theorem turns every finitely generated abelian group into explicit free and prime-power building blocks. It is a central classification result linking group theory, modules over principal ideal domains, and arithmetic.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's existence theorem: a finitely generated additive commutative group is additively equivalent to a finite-rank free integer part times a finite direct sum of cyclic prime-power groups. This declaration does not assert uniqueness, canonical factors, invariant-factor ordering, or an algorithm for computing the decomposition.

The selected declaration proves existence of a decomposition, not uniqueness or canonicity.
It does not order, merge, or require distinct prime-power factors.
The formal endpoint is an additive equivalence, not literal definitional equality of types.
It provides no algorithm for extracting the factors from a presentation.
ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics.

Source and local evidence

Where the theorem comes from

Existing declaration: AddCommGroup.equiv_free_prod_directSum_zmod in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.finitely-generated-abelian-groups.v001
Source: Open the pinned upstream reference