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.
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The symmetric group has the Coxeter presentation

Statement

For $n \geq 2$ , the symmetric group $S_{n}$ has the presentation

$S_{n} = ⟨ s_{1}, \dots, s_{n - 1} ∣ s_{i}^{2} = 1, s_{i} s_{i + 1} s_{i} = s_{i + 1} s_{i} s_{i + 1}, s_{i} s_{j} = s_{j} s_{i} (∣ i - j ∣ > 1) ⟩,$

where, after relabelling the underlying set $n = {0, \dots, n - 1}$ as ${1, \dots, n}$ , $s_{i}$ corresponds to the adjacent transposition $(i i + 1)$ . For $n = 0, 1$ , the trivial group has the empty presentation.

Facts & Assumptions

Given: The symmetric group $S_{n}$ and the adjacent transpositions $τ_{i} = (i i + 1)$ .

[L1]

The group $S_{n}$ is defined on $n = {0, \dots, n - 1}$ ; conjugating by the order-preserving bijection $k \mapsto k + 1$ identifies it with the conventional symmetric group on ${1, \dots, n}$ and transports adjacent transpositions (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation).

[F1]

Muger states in Section 4 that the symmetric groups have the presentation $S_{n} = ⟨ σ_{1}, \dots, σ_{n - 1} ∣ σ_{i}^{2} = 1, σ_{i} σ_{i + 1} σ_{i} = σ_{i + 1} σ_{i} σ_{i + 1}, σ_{i} σ_{j} = σ_{j} σ_{i} (∣ i - j ∣ > 1) ⟩ .$

Proof

technique · direct

1.1

For $n \geq 2$ , [F1] is exactly the displayed presentation on the conventional labels $1, \dots, n$ , and [L1] transports those generators to permutations of the library's underlying set $n$ .

F1L1

2.1

For $n = 0$ and $n = 1$ , there are no adjacent transpositions and $S_{n}$ is the trivial group, so the empty presentation applies.

L1algebra∎

Depends on

Used by

Symmetric coherence Theorem
The braid group surjects onto the symmetric group Theorem

Dependency tree · two levels

20 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)