CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

For a trivial action, first cohomology is Hom

Statement

If $G$ acts trivially on an abelian group $A$ , then

$H^{1} (G, A) ≅ Hom (G, A) .$

Facts & Assumptions

Given: A trivial action of $G$ on an abelian group $A$ .

[L1]

The crossed-homomorphism model computes $H^{1} (G, A)$ (The inhomogeneous one-cocycle model agrees with crossed homomorphisms in degree one).

[L2]

A group homomorphism $f : G \to A$ is characterized by $f (g h) = f (g) + f (h)$ in additive notation (Monoid homomorphism and group homomorphism).

Proof

technique · direct

1.1

Under the trivial action, the crossed-homomorphism identity becomes $z (g h) = z (g) + z (h)$ , which is exactly the homomorphism law of [L2]. So crossed homomorphisms are precisely the homomorphisms $G \to A$ .

givenL1L2

2.1

Principal crossed homomorphisms are all zero, because $g \cdot a - a = a - a = 0$ for every $g$ and $a$ . Therefore the quotient $H^{1} (G, A)$ is just the group of homomorphisms from step 1.1.

L1step 1.1algebra∎

Depends on

Used by

With trivial action, H¹(C₂,C₃) is zero Example

Dependency tree · two levels

8 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)