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

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Nonabelian first cohomology classifies complements as a pointed set

Statement

Let $G$ act on a group $M$ . Then the pointed set $H_{n a b}^{1} (G, M)$ is in canonical bijection with the $M$ -conjugacy classes of complements to the canonical copy of $M$ in $M ⋊ G$ , with the basepoint corresponding to the canonical complement ${(1, g) : g \in G}$ .

Facts & Assumptions

Given: An action of $G$ on a group $M$ .

[L1]

A graph subset is a complement exactly when its defining map is a crossed homomorphism (A graph subgroup is a complement exactly for a crossed homomorphism).

[L2]

Nonabelian first cohomology is the orbit set of crossed homomorphisms under the action $(a * z) (g) = a z (g) (g \cdot a)^{- 1}$ (First nonabelian cohomology as a pointed set).

Proof

technique · direct

1.1

For a nonabelian crossed homomorphism $z$ , the graph $Γ_{z}$ is a complement by [L1]. Conversely, every complement gives a unique graph map by the same lemma. So complements correspond exactly to nonabelian crossed homomorphisms.

givenL1

2.1

Conjugating $(z (g), g)$ by $(a, 1)$ in $M ⋊ G$ gives $(a, 1) (z (g), g) (a, 1)^{- 1} = (a z (g) (g \cdot a)^{- 1}, g)$ , which is the graph of the cocycle $(a * z) (g)$ from [L2]. Hence $M$ -conjugate complements correspond exactly to $M$ -orbits of cocycles.

L2step 1.1algebra

3.1

The trivial cocycle has graph ${(1, g) : g \in G}$ , the canonical complement to $M$ . Therefore the bijection of step 2.1 respects the distinguished basepoint and is an isomorphism of pointed sets.

L1L2step 2.1∎

Depends on

Used by

Dependency tree · two levels

6 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

Chaoli Li, Class field theory: proofs (standard reference, not scraped)
David A. Craven, Finite Group Theory (standard reference, not scraped)