CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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Nonuniqueness of Schur covers

Statement refuted

Every finite group has a unique Schur covering group up to isomorphism.

Counterexample

Assume the Axiom of Choice and supplied projective-resolution data for group homology. The groups $D_{8}$ and $Q_{8}$ are nonisomorphic Schur covers of $C_{2} \times C_{2}$ .

Given: The stated choice and resolution hypotheses. Both $D_{8}$ and $Q_{8}$ have central commutator subgroup of order two and quotient $C_{2} \times C_{2}$ .

1.1given

The exterior-square calculation gives $M (C_{2} \times C_{2}) ≅ C_{2}$ , so both are Schur covers.

2.1step 1.1algebra∎

$D_{8}$ has five involutions while $Q_{8}$ has one, hence they are not isomorphic.

Depends on

Used by

Nothing in the library uses this result yet.

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

Y. Bazlov and A. Berenstein, Cocycle twists and extensions of braided doubles, Sections 2.2–2.3 (standard reference, not scraped)
Clara Löh, Group Cohomology (standard reference, not scraped)