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: some extraspecial $p$ -group has order $p^{2 n}$

Statement refuted

some extraspecial $p$ -group has order $p^{2 n}$ .

Facts & Assumptions

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

[L1]

Every extraspecial $p$ -group is an internal central product of nonabelian subgroups of order $p^{3}$ pairwise intersecting in its centre (Every extraspecial $p$ -group is an internal central product of nonabelian subgroups of order $p^{3}$ ).

[L2]

An extraspecial $p$ -group has order $p^{1 + 2 n}$ for some $n \geq 1$ (An extraspecial $p$ -group has order $p^{1 + 2 n}$ for some $n \geq 1$ ).

[L3]

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G \approx n$ for some $n \in N$ . (The order $∣ G ∣$ of a finite group and the order $ord (g)$ of an element, with $ord (g) = \infty$ when no positive power of $g$ is the identity).

Refutation

technique · contradiction

1.1

The claim asserts that some extraspecial group has order $p^{2 n}$ .

L3assume-contra

2.1

The central-product decomposition forces the order to be $p$ times an even power of $p$ , so the exponent of the order is odd.

L1L2step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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: some extraspecial pp-group has order p2np2n