CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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 inversion crossed homomorphism C_2 → Z is not an ordinary homomorphism

Statement refuted

Every crossed homomorphism is an ordinary homomorphism.

Let the nontrivial element of $C_{2}$ act on $Z$ by negation. Then the map $z (1) = 0$ , $z (t) = 1$ is a crossed homomorphism but not a homomorphism.

Facts & Assumptions

Given: The false claim of FALSE: every crossed homomorphism is an ordinary homomorphism.

[L1]

The crossed-homomorphism identity for abelian coefficients is $z (g h) = z (g) + g \cdot z (h)$ (FALSE: every crossed homomorphism is an ordinary homomorphism).

Counterexample

technique · direct

1.1

With the negation action, $z (t^{2}) = 0 = 1 + (- 1) = z (t) + t \cdot z (t)$ , so $z$ is a crossed homomorphism.

givenL1algebra

2.1

But $z (t) = 1$ has infinite order in $Z$ , so $z$ cannot be an ordinary homomorphism from the order-two group $C_{2}$ . Therefore the displayed map is a counterexample to the claim.

L1step 1.1∎

Depends on

FALSE: every crossed homomorphism is an ordinary homomorphism

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

David A. Craven, Finite Group Theory (standard reference, not scraped)