TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Multiplier of an abelian group

Statement

For every abelian group $A$ , there is a natural isomorphism $M (A) ≅ ⋀^{2} A$ .

Facts & Assumptions

Given: Choose a free presentation $A = F / R$ ; since $A$ is abelian, $[F, F] \leq R$ .

Proof

technique · direct

1.1

The rule $\overset{ˉ}{x} \land \overset{ˉ}{y} \mapsto [x, y] [F, R]$ is independent of the chosen lifts: changing a lift by an element of $R$ changes the commutator by an element of $[F, R]$ . It is alternating and bilinear modulo $[F, R]$ , so the universal property gives a homomorphism $⋀^{2} A \to [F, F] / [F, R]$ .

givenalgebra

2.1

Conversely, the commutator quotient $[F, F] / [F, R]$ is generated by the classes of $[x, y]$ , subject exactly to the alternating bilinear commutator relations; sending such a class to $\overset{ˉ}{x} \land \overset{ˉ}{y}$ is therefore a well-defined inverse. Hopf's formula gives $M (A) = [F, F] / [F, R] ≅ ⋀^{2} A$ , and the construction is natural in $A$ .

step 1.1algebra∎

Depends on

Used by

Nonuniqueness of Schur covers Counterexample
Multiplier of a finite abelian group Example
Universal coefficients split naturally False statement

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

Clara Löh, Group Cohomology (standard reference, not scraped)