TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The braid group surjects onto the symmetric group

Statement

For $n \geq 2$ , use the one-based model $S_{n}^{(1)} : = Sym ({1, \dots, n})$ , identified with the library's $S_{n} = Sym ({0, \dots, n - 1})$ by conjugation with $κ_{n} (k) = k + 1$ (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation, Adjacent transpositions generate the finite symmetric group $S_{n}$ ). Write $S_{n}$ in this transported model below. The assignment $σ_{i} \mapsto (i i + 1)$ extends to a surjective homomorphism

$π_{n} : B_{n} ⟶ S_{n} .$

In the library's zero-based model, the same map is $σ_{i} \mapsto (i - 1 i)$ , obtained by conjugating $π_{n}$ with $κ_{n}^{- 1}$ .

Facts & Assumptions

Given: The Artin presentation of $B_{n}$ and the adjacent transpositions in the one-based model fixed in the Statement.

[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 $S_{n}$ (Adjacent transpositions generate the finite symmetric group $S_{n}$ ).

[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

1.1L1L3L4givenconstruct

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 $π_{n} : B_{n} \to S_{n}$ .

2.1L2step 1.1∎

The image of $π_{n}$ contains every adjacent transposition in the one-based model, so the transported generation clause of [L2] makes $π_{n}$ surjective. Conjugating its values by $κ_{n}^{- 1}$ preserves multiplication and surjectivity and sends $(i i + 1)$ to $(i - 1 i)$ , giving the stated zero-based version.

Depends on

Used by

The induced permutation does not determine a braid Counterexample
Specializing Burau at t = 1 recovers permutation data Example
The braid group on three strands and its quotient to S₃ Example
An involutive Yang–Baxter operator factors through the symmetric group Proposition
The symmetric and braided coherence theorems compare Sₙ with Bₙ Remark

Dependency tree · two levels

19 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)