CounterexampleConstruction: AI-adaptedVerification: 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.

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 $s r \neq r s$ (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 · two levels

18 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

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

The square-symmetry group has order 2323 and is nonabelian, so the order-p2p2 theorem does not extend to order p3p3