ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $|p^iG/p^{i+1}G|=p^{d_i}$ are $5,3,3,2,0.$

Facts & Assumptions

Given: The objects and hypotheses in the example.

[L1]

Suppose $G\cong\prod_{j<r}C_{p^{e_j}}$ with $e_j\ge1$ , and in additive notation write $p^iG=\{p^ig:g\in G\}$ . Define $d_i$ by $|p^iG/p^{i+1}G|=p^{d_i}$ . Then $d_i=|\{j:e_j\ge i+1\}|.$ Consequently, for every $k\ge1$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 55 results over 15 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)