Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Multiplier of a finitely presented group
Statement
Assume the Axiom of Choice and supplied projective-resolution data for group homology. A finitely presented group has finitely generated Schur multiplier.
Facts & Assumptions
Given: The stated choice and resolution hypotheses. Let have finitely many generators and finitely many defining relators. Choice implies the dependent-choice premise of Hopf formula for the Schur multiplier.
Proof
Modulo , every conjugate of a defining relator has the same class as that relator. Since is the normal closure of the finite relator set, their images generate . This quotient is abelian because .
Hopf's formula, under the stated hypotheses, identifies with the subgroup of this finitely generated abelian group. Subgroups of finitely generated abelian groups are finitely generated, so is finitely generated.
Depends on
Used by
Dependency tree · two levels
13 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)