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The fundamental theorem of finitely generated abelian groups from PID modules

Statement

Every finitely generated abelian group $G$ has unique canonical decompositions

$G ≅ Z^{r} \oplus Z / (n_{1}) \oplus \dots \oplus Z / (n_{t}), 1 < n_{1} ∣ \dots ∣ n_{t},$

and

$G ≅ Z^{r} \oplus ⨁_{p, j} Z / (p^{e_{p, j}}),$

where $r$ and the finite torsion data are unique up to the stated order. The torsion summand is finite. The trivial group has $r = 0$ and empty torsion data.

Facts & Assumptions

Given: The PID definition of Principal ideal domain.

[L1]

Abelian groups and $Z$ -modules have the same objects and morphisms; their subgroups, generated subobjects, cyclic objects, finite generation, and quotients agree (Abelian groups and $Z$ -modules have the same objects and morphisms).

[L2]

The integers form a commutative ring with multiplicative identity (The integers form a commutative ring).

[L3]

If $x, y \in Z$ are nonzero, then $x y \neq 0$ (The integers have no zero divisors; multiplicative cancellation).

[L4]

Every subgroup of $(Z, +)$ is cyclic (Every subgroup of $(Z, +)$ is $⟨ n ⟩ = n Z$ for exactly one natural number $n$ ).

[L5]

A finitely generated PID module is classified by its free rank and invariant factors, equivalently its elementary divisors: the free rank is unique, invariant factors are unique up to associates in their divisibility order, elementary divisors are unique up to associates and permutation, and two such modules are isomorphic exactly when these data agree (Uniqueness of invariant factors and elementary divisors over a PID).

[L6]

Every finitely generated module over a PID $R$ is isomorphic to $R^{s} \oplus R / (a_{1}) \oplus \dots \oplus R / (a_{t})$ with each $a_{i}$ a nonzero nonunit and $a_{1} ∣ \dots ∣ a_{t}$ , and also to $R^{r} \oplus ⨁_{p, j} R / (p^{e_{p, j}})$ with finitely many nonzero prime-power summands (Invariant-factor decomposition of a finitely generated module over a PID, Primary decomposition and elementary-divisor form for finitely generated PID modules).

Proof

technique · constructive

1.1L2L3

By [L2] and [L3], $Z$ is a commutative unital ring without zero divisors, hence an integral domain.

1.2L4algebra

Every ideal of $Z$ is an additive subgroup and is cyclic by [L4], so it is generated by one integer.

2.1step 1.1step 1.2given

Steps 1.1 and 1.2 verify the domain and principal-ideal clauses, so $Z$ is a PID.

3.1step 2.1L1L5L6construct

Regard $G$ as the canonical finitely generated $Z$ -module from [L1]. Applying [L6] over the PID in step 2.1 gives the free summand and the invariant-factor and elementary-divisor decompositions, and [L5] makes the free rank and the torsion data unique; [L1] translates the cyclic quotients back to cyclic abelian groups. Uniqueness and the converse construction are preserved by the dictionary.

4.1step 3.1algebradischarge-construct∎

Each torsion summand $Z / (n)$ is finite, and only finitely many occur, so their direct sum is finite. The free rank may be zero, and empty torsion data gives a finitely generated free abelian group.

The fundamental theorem of finitely generated abelian groups from PID modules

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources