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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Principal crossed homomorphisms form a subgroup

Statement

For an abelian $G$ -group $A$ , the set $B^{1} (G, A)$ of principal crossed homomorphisms is a subgroup of $Z^{1} (G, A)$ .

Facts & Assumptions

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

[L1]

Crossed homomorphisms with abelian coefficients form an abelian group (With abelian coefficients, crossed homomorphisms form an abelian group).

[L2]

Principal crossed homomorphisms are the maps $g \mapsto g \cdot a - a$ (Principal crossed homomorphism for abelian coefficients).

Proof

technique · direct

1.1

For $a \in A$ and $g, h \in G$ , $(d a) (g h) = g h \cdot a - a = g \cdot (h \cdot a - a) + (g \cdot a - a)$ , so every principal crossed homomorphism is a crossed homomorphism. Thus $B^{1} (G, A) \subseteq Z^{1} (G, A)$ by [L1] and [L2].

givenL1L2algebra

2.1

If $d a$ and $d b$ are principal, then $(d a + d b) (g) = g \cdot (a + b) - (a + b) = d (a + b) (g)$ and $- (d a) (g) = g \cdot (- a) - (- a) = d (- a) (g)$ . So $B^{1} (G, A)$ is closed under sums and inverses, hence is a subgroup of $Z^{1} (G, A)$ .

L1L2step 1.1algebra∎

Depends on

Used by

First cohomology via crossed homomorphisms Definition

Dependency tree · two levels

3 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)