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LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04
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With abelian coefficients, crossed homomorphisms form an abelian group

Statement

Let G act on an abelian group A by automorphisms. The set

Z1(G,A):={z:G→A:z(gh)=z(g)+g⋅z(h)}

is an abelian group under pointwise addition.

Facts & Assumptions

Given: A group G acting on an abelian group A.

[L1]

For abelian coefficients, a crossed homomorphism satisfies z(gh)=z(g)+g⋅z(h) (Crossed homomorphism for a G-group).

Proof

technique · direct
1.1

If z,w∈Z1(G,A), then (z+w)(gh)=z(gh)+w(gh)=z(g)+w(g)+g⋅z(h)+g⋅w(h)=(z+w)(g)+g⋅(z+w)(h), so z+w is again a crossed homomorphism by [L1].

givenL1algebra
2.1

The zero map is a crossed homomorphism, and if z∈Z1(G,A) then (−z)(gh)=−z(gh)=−z(g)+g⋅(−z(h)), so −z is one as well. Thus pointwise addition makes Z1(G,A) a subgroup of the abelian group AG. In particular it is abelian.

L1step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

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Sources