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  "body": "\n## Definition\n\nLet $I$ be a set.\nA *Coxeter matrix* on $I$ is a family $M = (m_{ij})_{i, j \\in I}$ with values in $\\set{1, 2, 3, \\ldots} \\cup \\set{\\infty}$ such that:\n\n1. $m_{ii} = 1$ for all $i \\in I$;\n2. $m_{ij} = m_{ji}$ for all $i, j \\in I$;\n3. $m_{ij} \\ge 2$ for all $i \\ne j$.\n\nThe *Coxeter group* of $M$ is the group with the [[presentation|0JGEQXQ4AC]]\n$$W(M) = \\langle s_i \\ (i \\in I) \\mid (s_i s_j)^{m_{ij}} = 1 \\text{ for all } i, j \\in I \\text{ with } m_{ij} \u003c \\infty \\rangle.$$\nA group is a *Coxeter group* if it is [[isomorphic|08MG7H8JW5]] to $W(M)$ for some Coxeter matrix $M$.\nThe pair $(W(M), \\set{s_i})$ is a *Coxeter system*.\n\n## Remarks\n\nThe relation for $i = j$ says $s_i^2 = 1$.\nFor $m_{ij} = 2$ the relation says that $s_i$ and $s_j$ commute.\nIn $W(M)$ each $s_i$ has order exactly $2$, and $s_i s_j$ has order exactly $m_{ij}$.\nThe matrix is often drawn as a Coxeter diagram: a vertex for each $i$, and an edge labelled $m_{ij}$ when $m_{ij} \\ge 3$.\n\nExample: the [[symmetric group|0515A6D2Y1]] $S_n$ is a Coxeter group.\nTake $s_i = (i \\ i{+}1)$ for $1 \\le i \\le n - 1$, with $m_{i, i+1} = 3$ and $m_{ij} = 2$ for $\\abs{i - j} \\ge 2$.\nThe relations hold: $(i \\ i{+}1)(i{+}1 \\ i{+}2)$ is a $3$-cycle, and disjoint transpositions commute.\nA theorem of Moore says that these relations define $S_n$.\n\nNon-example: the cyclic group $\\Z/4$ is not a Coxeter group.\nA Coxeter group is generated by elements of order $2$, so in its abelianization every element has order at most $2$.\nBut $\\Z/4$ is abelian, so it equals its abelianization, and it has an element of order $4$.\n",
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