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: the centre of an extraspecial $p$ -group has a complement

Statement refuted

the centre of an extraspecial $p$ -group has a complement.

Facts & Assumptions

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

[L1]

The centre of an extraspecial $p$ -group has no complement (The centre of an extraspecial $p$ -group has no complement).

[L2]

In this situation $H$ is called a complement to $N$ in $G$ . (An internal semidirect product and a complement to a normal subgroup).

Refutation

technique · contradiction

1.1

The claim asserts the existence of a subgroup meeting the centre trivially and multiplying with it to the whole group.

L2assume-contra

2.1

Such a subgroup would be isomorphic to the elementary abelian central quotient, hence abelian, forcing the whole group to be abelian.

L1L2step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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: the centre of an extraspecial pp-group has a complement