ExampleConstruction: 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.

Crossed homomorphisms from a cyclic group are determined by the value on a generator

Example

Let $G = ⟨ t ⟩$ be cyclic and let $M$ be a $G$ -group. A crossed homomorphism $z : G \to M$ is determined by the single value $z (t)$ .

Facts & Assumptions

Given: A cyclic group $G = ⟨ t ⟩$ acting on a group $M$ , and a crossed homomorphism $z : G \to M$ .

[L1]

Crossed homomorphisms satisfy $z (g h) = z (g) (g \cdot z (h))$ (Crossed homomorphism for a G-group).

Verification

technique · direct

1.1givenL1algebra

Repeatedly applying [L1] gives $z (t^{n}) = z (t) (t \cdot z (t)) \dots (t^{n - 1} \cdot z (t))$ for every integer $n \geq 1$ . So all positive powers of $t$ are determined by $z (t)$ .

2.1L1step 1.1algebra∎

Since $1 = z (1) = z (t^{n} t^{- n}) = z (t^{n}) (t^{n} \cdot z (t^{- n}))$ , the value on $t^{- n}$ is determined by the already known value on $t^{n}$ . Therefore every value of $z$ is determined by $z (t)$ .

Depends on

Crossed homomorphism for a G-group

Used by

The affine group AGL(1,p) has one kernel-conjugacy class of complements to its translation subgroup Example

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)