CorollaryStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-26 (gpt-6-sol)

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Central extensions are classified by H^2 with trivial action

Statement

Assume the Axiom of Choice (The Axiom of Choice). If $A$ is an abelian group with trivial $G$ -action, then equivalence classes of central extensions

$1 \to A \to E \to G \to 1$

are classified by $H^{2} (G, A)$ .

Facts & Assumptions

Given: The Axiom of Choice, a group $G$ , and an abelian group $A$ with trivial $G$ -action.

[A1]

Choice supplies the normalized sections required by [L1] for arbitrary quotient surjections.

[L1]

$H^{2} (G, M)$ classifies extensions with a fixed abelian kernel action (H^2 classifies extensions with fixed abelian kernel action).

[L2]

The zero class is the split semidirect-product class (The zero H^2 class is equivalent to splitting).

Proof

technique · direct

1.1givenalgebra

With trivial action, the condition defining an extension inducing the prescribed action says that every lift of every $g \in G$ centralizes the kernel $A$ . That is exactly the statement that the kernel is central in $E$ .

2.1A1L1L2step 1.1

Under [A1], [L1] applies with $M = A$ and identifies $H^{2} (G, A)$ with the equivalence classes of central extensions. The split class singled out by [L2] is the direct-product class because the action is trivial.

3.1step 2.1∎

Hence central extensions are classified by $H^{2} (G, A)$ .

Depends on

Used by

Central extensions of a cyclic group Example
Quaternion and dihedral central extension classes Example

Dependency tree · two levels

7 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

Clara Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)