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✓ 6 results · all verified · 6 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs; all 6 also cleared it.

Schur Multipliers and Universal Central Extensions — Examples

1 · Prerequisites

2 · Summary

The examples distinguish a universal central extension, which requires a perfect base, from finite Schur covers, which need not be unique.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06Open item page →

Multiplier of a cyclic group

Example

$M (C) = 0$ for every cyclic group $C$ .

Facts & Assumptions

Given: Let $C$ be cyclic.

Verification

technique · direct

1.1

Multiplier of a cyclic group applies to every finite or infinite cyclic group and gives $M (C) = 0$ .

given

2.1

This is exactly the claimed cyclic-group calculation.

step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06Open item page →

Multiplier of a finite abelian group

Example

For $A ≅ C_{n_{1}} \times \dots \times C_{n_{r}}$ , $M (A) ≅ ⨁_{1 \leq i < j \leq r} C_{\gcd (n_{i}, n_{j})} .$

Facts & Assumptions

Given: Use the invariant-factor decomposition of $A$ .

Verification

technique · direct

1.1

Exterior squares take finite direct sums to the sum of the exterior squares of the summands and the pairwise tensor products. Each cyclic summand has zero exterior square, while $C_{m} \otimes C_{n} ≅ C_{\gcd (m, n)}$ .

givenalgebra

2.1

Applying Multiplier of an abelian group gives the displayed formula, including the empty sum $0$ when $r \leq 1$ .

step 1.1algebra∎

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06Open item page →

Binary icosahedral cover of A5

Example

$SL (2, 5) \to PSL (2, 5) ≅ A_{5}$ is the sourced binary-icosahedral universal-cover example.

Facts & Assumptions

Given: Use Weibel, §6.9, Example 6.9.1, cited above.

[L1]

Weibel's Example 6.9.1 identifies $SL (2, 5) \to PSL (2, 5)$ as a universal central extension with kernel ${\pm I}$ .

Verification

technique · direct

1.1

The central quotient map $SL (2, 5) \to PSL (2, 5)$ has kernel ${\pm I} ≅ C_{2}$ , and the cited example identifies it as the universal central extension.

L1given

2.1

With the standard isomorphism $PSL (2, 5) ≅ A_{5}$ , this is the binary-icosahedral universal cover asserted in the statement.

step 1.1algebra∎

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06Open item page →

Hopf formula from a one-relator presentation

Example

For ⟨x | x^n⟩, Hopf’s quotient is trivial.

Verification

Given: Take $F = ⟨ x ⟩$ and $R = ⟨ x^{n} ⟩$ .

1.1

As $F$ is cyclic, $[F, F] = 1$ and $R \cap [F, F] = 1$ .

given

2.1

Thus Hopf’s quotient is trivial.

step 1.1algebra∎

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06Open item page →

A stem extension that is not universal

Example

The central extension C2→D8→C2×C2 is stem but not universal because its base is not perfect.

Verification

Given: The center and commutator subgroup of $D_{8}$ are both $⟨ r^{2} ⟩$ .

1.1

Thus $C_{2} \to D_{8} \to C_{2} \times C_{2}$ is stem.

given

2.1

Its nontrivial abelian base is not perfect, so it cannot be universal.

step 1.1algebra∎

CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06Open item page →

Nonuniqueness of Schur covers

Statement refuted

D8 and Q8 are nonisomorphic Schur covers of C2×C2.

Counterexample

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

1.1

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

given

2.1

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

step 1.1algebra∎

Schur Multipliers and Universal Central Extensions — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Multiplier of a cyclic group

Example

Facts & Assumptions

Verification

Multiplier of a finite abelian group

Example

Facts & Assumptions

Verification

Binary icosahedral cover of A5

Example

Facts & Assumptions

Verification

Hopf formula from a one-relator presentation

Example

Verification

A stem extension that is not universal

Example

Verification

Nonuniqueness of Schur covers

Statement refuted

Counterexample

Sources