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

How statement and proof provenance work

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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.

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Kernel-conjugate complements differ by a principal crossed homomorphism

Example

Let $C_{2} = ⟨ t ⟩$ act on an abelian group $A$ . If $z : C_{2} \to A$ is a crossed homomorphism and $a \in A$ , then the complements $Γ_{z}$ and $Γ_{z_{a}}$ are conjugate by a kernel element, where $z_{a} (g) = z (g) + g \cdot a - a (g \in C_{2}) .$

Facts & Assumptions

Given: An action of $C_{2}$ on an abelian group $A$ , a crossed homomorphism $z$ , and an element $a \in A$ .

[L1]

Conjugating by a kernel element changes the crossed homomorphism by a principal one (Kernel conjugation by an element of the coefficient group corresponds to a principal crossed homomorphism).

[L2]

First cohomology identifies kernel-conjugacy classes of complements (First cohomology classifies complements up to kernel conjugacy).

Verification

technique · direct

1.1givenL1construct

Define $z_{a}$ on every $g \in C_{2}$ by the displayed formula. Fact [L1] gives $(- a, 1) Γ_{z} (a, 1) = Γ_{z_{a}}$ . So the two complements are conjugate by the kernel element $(- a, 1)$ .

2.1L1L2step 1.1∎

Therefore they determine the same class under the bijection of [L2], exactly because their defining cocycles differ by the principal crossed homomorphism attached to $a$ .

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

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