ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The braid group on three strands and its quotient to S_3

Example

The three-strand braid group has presentation

$B_{3} = ⟨ σ_{1}, σ_{2} ∣ σ_{1} σ_{2} σ_{1} = σ_{2} σ_{1} σ_{2} ⟩,$

and the quotient map to $S_{3}$ sends $σ_{1}$ to $(1 2)$ and $σ_{2}$ to $(2 3)$ .

Facts & Assumptions

Given: The Artin presentation of braid groups and the standard quotient to the symmetric group.

[L1]

The Artin presentation defines $B_{n}$ (The braid group by Artin presentation).

[L2]

The map $σ_{i} \mapsto (i i + 1)$ extends to a surjection $B_{n} \to S_{n}$ (The braid group surjects onto the symmetric group).

Verification

technique · direct

1.1

Specializing [L1] to $n = 3$ leaves two generators and only one braid relation, because there is no pair $i, j$ with $∣ i - j ∣ > 1$ among ${1, 2}$ . This gives the displayed presentation of $B_{3}$ .

L1givenalgebra

2.1

Specializing [L2] to $n = 3$ gives the quotient map $B_{3} \to S_{3}$ with $σ_{1} \mapsto (1 2)$ and $σ_{2} \mapsto (2 3)$ .

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Michael Muger, Tensor Categories: A Selective Guided Tour, Section 4 (standard reference, not scraped)