TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passverified 2026-09-26 (gpt-6-sol)

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The two-strand braid group is infinite cyclic

Statement

The braid group $B_{2}$ is cyclic, generated by $σ_{1}$ , and $σ_{1}$ has infinite order. Hence $B_{2}$ is an infinite cyclic group.

Facts & Assumptions

Given: The Artin presentation of the braid groups.

[L1]

The group $B_{2}$ is presented by one generator $σ_{1}$ and no braid relations (The braid group by Artin presentation).

[L2]

In the free group on an alphabet, reduced words are the elements, the empty word is the identity, and two reduced words represent the same element only when they are identical (Reduced words form the free group on an alphabet).

Proof

technique · direct

1.1givenL1algebra

By [L1], the presentation of $B_{2}$ has one generator and no defining relator, so every element of $B_{2}$ is a power of $σ_{1}$ . Thus $B_{2}$ is cyclic.

2.1L1L2step 1.1

With no relators present, the presented group is the free group on the singleton alphabet ${σ_{1}}$ . For $m > 0$ the power $σ_{1}^{m}$ is represented by a word of $m$ copies of $σ_{1}$ ; for $m < 0$ it is represented by $∣ m ∣$ copies of the formal inverse. Each is a nonempty reduced word, distinct from the empty identity word by [L2]. Hence $σ_{1}^{m} \neq 1$ for every nonzero integer $m$ .

3.1step 1.1step 2.1∎

Therefore the cyclic group $B_{2} = ⟨ σ_{1} ⟩$ is infinite, so it is infinite cyclic.

Depends on

Used by

The braid category is braided but not symmetric Counterexample
The two-strand braiding in the braid category has infinite order Example
Braided coherence fails in the symmetric form Theorem
The reduced Burau representation is faithful for at most three strands Theorem

Dependency tree · two levels

9 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)