TheoremStatement: Literature-sourcedProof: 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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The braid group surjects onto the symmetric group

Statement

For every $n \geq 2$ , the assignment $σ_{i} \mapsto (i i + 1)$ extends to a surjective homomorphism

$π_{n} : B_{n} ⟶ S_{n} .$

Facts & Assumptions

Given: The Artin presentation of $B_{n}$ and the adjacent transpositions in $S_{n}$ .

[L1]

The braid group $B_{n}$ has generators $σ_{1}, \dots, σ_{n - 1}$ with the Artin braid and distant-commutativity relations (The braid group by Artin presentation).

[L2]

The adjacent transpositions generate $S_{n}$ (The adjacent transpositions $(1 2), (2 3), \dots, (n - 1 n)$ generate $S_{n}$ ).

[L3]

The symmetric group satisfies the Coxeter relations for adjacent transpositions (The symmetric group has the Coxeter presentation).

[L4]

A map of generators satisfying the relators extends uniquely from a presented group (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group).

Proof

technique · direct

1.1

By [L3], the adjacent transpositions in $S_{n}$ satisfy the braid relations and the distant-commutativity relations from [L1]. Therefore [L4] extends the assignment $σ_{i} \mapsto (i i + 1)$ to a homomorphism $π_{n} : B_{n} \to S_{n}$ .

L1L3L4givenconstruct

2.1

The image of $π_{n}$ contains every adjacent transposition, so [L2] makes $π_{n}$ surjective.

L2step 1.1∎

Depends on

Used by

Dependency tree · two levels

18 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)