The Structure Theorem for Finitely Generated Abelian Groups

Every finitely generated abelian group is a direct sum of cyclic groups, completely classifying these groups up to isomorphism.

Statement

Theorem9.2Fundamental theorem of finitely generated abelian groups

Every finitely generated abelian group $G$ is isomorphic to:

$G \cong \mathbb{Z}^r \oplus \mathbb{Z}/d_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/d_s\mathbb{Z},$

where $r \geq 0$ (the rank or free rank) and $d_1 \mid d_2 \mid \cdots \mid d_s$ with each $d_i \geq 2$ (invariant factor form).

Equivalently, in elementary divisor form:

$G \cong \mathbb{Z}^r \oplus \mathbb{Z}/p_1^{e_1}\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/p_t^{e_t}\mathbb{Z},$

where each $p_i$ is prime and $e_i \geq 1$ . Both decompositions are unique.

Proof Sketch

Proof

This is the structure theorem for finitely generated modules over the PID $\mathbb{Z}$ .

Step 1: $G$ is a quotient of $\mathbb{Z}^n$ (by choosing generators). So $G \cong \mathbb{Z}^n / K$ where $K$ is a subgroup.

Step 2: By the theorem on subgroups of free abelian groups, $K \cong \mathbb{Z}^m$ for some $m \leq n$ (every subgroup of $\mathbb{Z}^n$ is free abelian).

Step 3: Choose bases $\{e_1, \ldots, e_n\}$ for $\mathbb{Z}^n$ and $\{f_1, \ldots, f_m\}$ for $K$ such that $f_i = d_i e_i$ for $i = 1, \ldots, m$ with $d_1 \mid d_2 \mid \cdots \mid d_m$ (this is the Smith normal form of the inclusion matrix).

Step 4: $G \cong \mathbb{Z}^n / \bigoplus d_i \mathbb{Z} \cong \bigoplus_{i=1}^{m} \mathbb{Z}/d_i\mathbb{Z} \oplus \mathbb{Z}^{n-m}$ . Dropping the $d_i = 1$ terms gives the invariant factor form. $\blacksquare$

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Classification

ExampleClassifying abelian groups of given order

All abelian groups of order 36 ( $= 2^2 \cdot 3^2$ ):

$\mathbb{Z}/4 \oplus \mathbb{Z}/9 \cong \mathbb{Z}/36$ (invariant factors: $36$ ).
$\mathbb{Z}/4 \oplus \mathbb{Z}/3 \oplus \mathbb{Z}/3$ (elementary divisors: $4, 3, 3$ ).
$\mathbb{Z}/2 \oplus \mathbb{Z}/2 \oplus \mathbb{Z}/9$ (elementary divisors: $2, 2, 9$ ).
$\mathbb{Z}/2 \oplus \mathbb{Z}/2 \oplus \mathbb{Z}/3 \oplus \mathbb{Z}/3$ (elementary divisors: $2, 2, 3, 3$ ).

Total: 4 groups (corresponding to partitions $2 = 2 = 1+1$ for $p=2$ and $2 = 2 = 1+1$ for $p=3$ : $2 \times 2 = 4$ ).

RemarkConnection to number theory

The structure theorem classifies: finite abelian groups up to isomorphism, class groups of number fields (when finitely generated), torsion subgroups of elliptic curves over finite fields, and homology groups in algebraic topology.