False statementConstruction: 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.

FALSE: every crossed homomorphism is an ordinary homomorphism

Statement

Every crossed homomorphism is an ordinary homomorphism.

Facts & Assumptions

Given: The nontrivial element $t$ of $C_{2}$ acting on $Z$ by $t \cdot n = - n$ .

[L1]

A crossed homomorphism satisfies $z (g h) = z (g) + g \cdot z (h)$ for abelian coefficients (Crossed homomorphism for a G-group).

Refutation

technique · direct

1.1

Define $z : C_{2} \to Z$ by $z (1) = 0$ and $z (t) = 1$ . Then $z (t^{2}) = z (1) = 0 = 1 + t \cdot 1 = z (t) + t \cdot z (t)$ , so [L1] shows that $z$ is a crossed homomorphism.

givenL1algebra

2.1

But $z$ is not an ordinary homomorphism, because a homomorphism $C_{2} \to Z$ must send $t$ to an element of order dividing $2$ , hence to $0$ , whereas $z (t) = 1$ . Therefore the claim is false.

L1step 1.1algebra∎

Depends on

Crossed homomorphism for a G-group

Used by

The inversion crossed homomorphism C₂ → Z is not an ordinary homomorphism Counterexample

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)