False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

FALSE: two extraspecial $p$ -groups whose commutator pairings agree are isomorphic

Statement refuted

two extraspecial $p$ -groups whose commutator pairings agree are isomorphic.

Facts & Assumptions

Given: The proposed claim together with the witness named in the Statement refuted.

[F1]

For an extraspecial $p$ -group $P$ with $Z (P) = ⟨ z ⟩$ , the commutator pairing is the map $b_{z} (\overset{ˉ}{x}, \overset{ˉ}{y}) \in Z / p$ determined by $[x, y] = z^{b_{z} (\overset{ˉ}{x}, \overset{ˉ}{y})}$ (The commutator pairing of an extraspecial $p$ -group relative to a chosen generator of its centre).

[L1]

The commutator pairings of $Dih (C_{4})$ and $Q_{8}$ agree, while the groups are not isomorphic (The commutator pairings of $Dih (C_{4})$ and $Q_{8}$ are the same, while the groups are not isomorphic).

Refutation

technique · contradiction

1.1

The claim asserts that equality of commutator pairings forces an isomorphism of groups.

F1assume-contra

2.1

The example [L1] gives two extraspecial groups of order eight with the same commutator pairing and different isomorphism type, contradicting the claim.

L1step 1.1discharge-contradiction∎

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

D. A. Craven, The Theory of p-Groups (Hilary Term 2008), 48 pp. (standard reference, not scraped)
M. van Beek, Topics in Finite p-Groups, 62 pp. (standard reference, not scraped)
D. Kaur and A. Kulshrestha, Characters of real special 2-groups (arXiv:1510.06583v1) (standard reference, not scraped)

FALSE: two extraspecial pp-groups whose commutator pairings agree are isomorphic