ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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, $C_{16},\quad C_8\times C_2,\quad C_4\times C_4,\quad C_4\times C_2\times C_2,\quad C_2^4.$

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 $(e_1,\ldots,e_r)$ with $e_1+\cdots+e_r=n$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 49 results over 17 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)