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  "body": "\n## Definition\n\nLet $n \\ge 1$ be an integer.\nThe *braid group on $n$ strands* is the [[group|G5S1M9]] $B_n$ with the [[presentation|0JGEQXQ4AC]] with generators $\\sigma_1, \\ldots, \\sigma_{n-1}$ and the relations\n\n1. $\\sigma_i \\sigma_j = \\sigma_j \\sigma_i$ for $1 \\le i, j \\le n - 1$ with $\\abs{i - j} \\ge 2$;\n2. $\\sigma_i \\sigma_{i+1} \\sigma_i = \\sigma_{i+1} \\sigma_i \\sigma_{i+1}$ for $1 \\le i \\le n - 2$.\n\n## Remarks\n\nGeometrically, $\\sigma_i$ is the braid in which strand $i$ crosses over strand $i + 1$ and the other strands go straight down.\nBy a theorem of Artin, this presentation gives the group of isotopy classes of geometric braids on $n$ strands.\nIt is the [[Artin group|SAWBDBHE41]] of the Coxeter matrix of $S_n$.\n\nExample: $B_1$ is trivial and $B_2 = \\langle \\sigma_1 \\mid \\ \\rangle \\cong \\Z$.\nThe group $B_3 = \\langle \\sigma_1, \\sigma_2 \\mid \\sigma_1 \\sigma_2 \\sigma_1 = \\sigma_2 \\sigma_1 \\sigma_2 \\rangle$ is not abelian.\nThe map $\\sigma_i \\mapsto (i \\ i{+}1)$ respects both relations, so it gives a surjective homomorphism $B_n \\to S_n$ onto the [[symmetric group|0515A6D2Y1]], and $S_3$ is not abelian.\nThe kernel of $B_n \\to S_n$ is the pure braid group.\n\nNon-example: the symmetric group $S_n$ with the transpositions $(i \\ i{+}1)$ satisfies relations 1 and 2, but it is not $B_n$ for $n \\ge 2$.\nIn $S_n$ the generators also satisfy $(i \\ i{+}1)^2 = 1$, while in $B_n$ each $\\sigma_i$ has infinite order, since the map $B_n \\to \\Z$ with $\\sigma_i \\mapsto 1$ respects the relations and sends $\\sigma_i^k$ to $k$.\n",
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    "name": "Braid group",
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