TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Fundamental theorem of finite abelian groups: invariant-factor form

Statement

For every finite abelian group $G$ there is a unique list $1<n_1\mid\cdots\mid n_r$ such that $G\cong C_{n_1}\times\cdots\times C_{n_r}$ . Moreover $|G|=n_1\cdots n_r$ . The trivial group corresponds to the empty list and empty product.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[L1]

Every finite abelian group is isomorphic to a finite direct product of cyclic groups of prime-power order. The multiset of their orders is uniquely determined by the group, up to permutation of the factors. (Fundamental theorem of finite abelian groups: elementary-divisor form).

[L2]

Every multiset of prime-power elementary divisors regroups in exactly one way into an invariant-factor list. (Elementary divisors regroup uniquely into invariant factors).

[L3]

An invariant-factor list for a finite abelian group $G$ is a finite list of integers $1<n_1\mid n_2\mid\cdots\mid n_r$ together with an isomorphism $G\cong C_{n_1}\times\cdots\times C_{n_r}$ . The cyclic factors and product use thm-classification-of-cyclic-groups and def-external-direct-product-of-groups. Unit factors are omitted. The trivial group has the empty list. (Invariant-factor data for a finite abelian group).

[L4]

If $G$ and $H$ are finite groups, then their external direct product is finite and has order $|G\times H|=|G|\,|H|$ . (For finite groups $G$ and $H$ , $|G\times H|=|G|\,|H|$ ).

Proof

technique · direct

1.1

The elementary-divisor theorem supplies a unique multiset of prime powers, and the regrouping lemma converts it into a unique invariant-factor list.

givenL1L2L3L4

2.1

The order formula for finite direct products gives $|G|=\prod_i n_i$ ; for the empty list this product is $1$ , the order of the trivial group.

step 1.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 77 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Keith Conrad, Decomposition of Finite Abelian Groups, §§1-4 (standard reference, not scraped)
Richard Elman, Lectures on Abstract Algebra, Ch. 14 (standard reference, not scraped)