False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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: every special $p$ -group is extraspecial

Statement refuted

every special $p$ -group is extraspecial.

Facts & Assumptions

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

[F1]

A finite $p$ -group $P$ is special when $Z (P) = P^{'} = Φ (P)$ is elementary abelian, and extraspecial when in addition $P$ is nonabelian and this common subgroup has order $p$ (Special and extraspecial $p$ -groups).

[L1]

$Z (G) : = {z \in G : z g = g z for every g \in G} .$ (The center $Z (G)$ of a group).

Refutation

technique · contradiction

1.1

The claim asserts that a special $p$ -group always has centre of order $p$ .

F1assume-contra

2.1

The direct product of two Heisenberg groups is special with centre of order $p^{2}$ , refuting the claim.

F1L1step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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: every special pp-group is extraspecial