CounterexampleConstruction: AI-adaptedVerification: 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.

The square-symmetry group has order $2^3$ and is nonabelian, so the order- $p^2$ theorem does not extend to order $p^3$

Statement refuted

False claim. Every group of order $p^3$ , for prime $p$ , is abelian.

Facts & Assumptions

Given: The square-symmetry group $D$ constructed in the preceding example.

[L1]

Every group of order $p^2$ is abelian (Every group of order $p^2$ , for prime $p$ , is abelian).

[L2]

The square-symmetry group has eight displayed elements and contains $r,s$ with $sr\ne rs$ (The square-symmetry group has class equation $8=2+2+2+2$ ).

Counterexample

technique · direct

1.1

By [L2], $D$ is nonabelian and has order $8=2^3$ .

L2algebra

2.1

Since $2$ is prime, $D$ refutes the false claim and shows that the exponent $2$ in [L1] cannot be replaced by $3$ .

step 1.1L1∎

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: 74 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

T. W. Judson, Abstract Algebra: Theory and Applications, 14.2 (standard reference, not scraped)

The square-symmetry group has order 232^323 and is nonabelian, so the order-p2p^2p2 theorem does not extend to order p3p^3p3