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  "body": "\n## Definition\n\nLet $I$ be a set and let $M = (m_{ij})_{i, j \\in I}$ be a Coxeter matrix, as in the definition of a [[Coxeter group|G6FND7X69J]].\nFor letters $x, y$ and an integer $m \\ge 2$, let $\\Pi(x, y; m) = x y x y \\cdots$ be the alternating word of length $m$ that begins with $x$.\nThe *Artin group* of $M$ is the group with the [[presentation|0JGEQXQ4AC]]\n$$A(M) = \\langle s_i \\ (i \\in I) \\mid \\Pi(s_i, s_j; m_{ij}) = \\Pi(s_j, s_i; m_{ij}) \\text{ for all } i \\ne j \\text{ with } m_{ij} \u003c \\infty \\rangle.$$\nA group is an *Artin group*, or *Artin–Tits group*, if it is [[isomorphic|08MG7H8JW5]] to $A(M)$ for some Coxeter matrix $M$.\n\n## Remarks\n\nFor $m_{ij} = 2$ the relation is $s_i s_j = s_j s_i$, and for $m_{ij} = 3$ it is $s_i s_j s_i = s_j s_i s_j$.\nThe quotient of $A(M)$ by the relations $s_i^2 = 1$ is the Coxeter group $W(M)$.\n$A(M)$ is of *finite type*, or spherical, when $W(M)$ is finite.\n\nExample: the [[braid group|BNR4H5GCTP]] $B_n$ is the Artin group with $I = \\set{1, \\ldots, n-1}$, $m_{i, i+1} = 3$ and $m_{ij} = 2$ for $\\abs{i - j} \\ge 2$.\nThese are exactly Artin's relations $\\sigma_i \\sigma_{i+1} \\sigma_i = \\sigma_{i+1} \\sigma_i \\sigma_{i+1}$ and $\\sigma_i \\sigma_j = \\sigma_j \\sigma_i$.\nIf all $m_{ij} = \\infty$, there are no relations and $A(M)$ is the [[free group|0KTW26C80J]] on $I$; if all $m_{ij} = 2$ for $i \\ne j$, then $A(M)$ is free abelian.\n\nNon-example: the group $\\Z/2$ is not an Artin group.\nIn the abelianization of $A(M)$, a relation with $m_{ij}$ even becomes trivial, and a relation with $m_{ij}$ odd becomes $s_i = s_j$.\nSo the abelianization of an Artin group is free abelian, but $\\Z/2$ is abelian and not free abelian.\n",
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