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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Successive p-multiple layers recover the summands of C_p^2 times C_{p^3} times C_{p^4}^2

Example

For $G = C_{p}^{2} \times C_{p^{3}} \times C_{p^{4}}^{2},$ the dimensions $d_{i}$ defined by $∣ p^{i} G / p^{i + 1} G ∣ = p^{d_{i}}$ are $5, 3, 3, 2, 0.$

Facts & Assumptions

Given: The objects and hypotheses in the example.

[L1]

Suppose $G ≅ \prod_{j < r} C_{p^{e_{j}}}$ with $e_{j} \geq 1$ , and in additive notation write $p^{i} G = {p^{i} g : g \in G}$ . Define $d_{i}$ by $∣ p^{i} G / p^{i + 1} G ∣ = p^{d_{i}}$ . Then $d_{i} = ∣ {j : e_{j} \geq i + 1} ∣ .$ Consequently, for every $k \geq 1$ , the number of summands of order $p^{k}$ is $d_{k - 1} - d_{k}$ , so the elementary divisors are intrinsic. (The successive quotients p^iG/p^{i+1}G recover the cyclic summand multiplicities of a finite abelian p-group).

[L2]

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 $H$ , $∣ G \times H ∣ = ∣ G ∣ ∣ H ∣$ ).

Verification

technique · direct

1.1

At layers $i = 0, 1, 2, 3, 4$ , the numbers of exponents among $1, 1, 3, 4, 4$ that exceed $i$ are respectively $5, 3, 3, 2, 0$ .

givenL1L2

2.1

The differences $d_{0} - d_{1} = 2$ , $d_{1} - d_{2} = 0$ , $d_{2} - d_{3} = 1$ , and $d_{3} - d_{4} = 2$ recover two $C_{p}$ factors, no $C_{p^{2}}$ factor, one $C_{p^{3}}$ factor, and two $C_{p^{4}}$ factors.

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)