Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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Hopf formula is obviously independent

Statement

The presentation-independence of Hopf's quotient follows directly from its displayed formula, without identifying it with M(G).

Facts & Assumptions

Given: Take two unrelated free presentations of the same group.

Refutation

technique · direct
1.1

Their groups (R∩[F,F])/[F,R] are built from different free groups and there is no presentation-free identification between the displayed quotients from their formulas alone.

given
2.1

Independence is obtained only after Hopf's theorem identifies each quotient with the invariant M(G), as in Hopf formula is presentation-independent. It is not a formal or "obvious" consequence of writing the quotient.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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