ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-11

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The five abelian groups of order sixteen

Example

The abelian groups of order $16$ are, up to isomorphism, $C_{16}, C_{8} \times C_{2}, C_{4} \times C_{4}, C_{4} \times C_{2} \times C_{2}, C_{2}^{4} .$

Facts & Assumptions

Given: The objects and hypotheses in the example.

[L1]

For a prime $p$ and $n > 0$ , isomorphism classes of abelian groups of order $p^{n}$ are in bijection with partitions of $n$ . For $n = 0$ , the unique group is the trivial group and corresponds separately to the empty partition. (Isomorphism classes of abelian groups of order p^n are counted by partitions of n).

[L2]

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).

[L3]

For $n > 0$ , a partition of $n$ is a finite nondecreasing list of positive integers $(e_{1}, \dots, e_{r})$ with $e_{1} + \dots + e_{r} = n$ , using finite natural sums as in def-nat-finite-sum-and-product and naturals as in def-natural-numbers. Equality is equality of these lists. The nondecreasing convention removes permutations from the data. (Partitions of a positive integer).

Verification

technique · direct

1.1

The partitions of $4$ are $4$ , $3 + 1$ , $2 + 2$ , $2 + 1 + 1$ , and $1 + 1 + 1 + 1$ .

givenL1L2L3

2.1

Replacing each part $e$ by $C_{2^{e}}$ gives the displayed groups. The partition bijection makes the list exhaustive and prevents repetitions.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)