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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] and let $S$ be a subset of $G$ that generates $G$.\nThe *word length* $\\abs{g}_S$ of $g \\in G$ is the least integer $n \\ge 0$ such that $g = s_1^{\\varepsilon_1} s_2^{\\varepsilon_2} \\cdots s_n^{\\varepsilon_n}$ for some $s_1, \\ldots, s_n \\in S$ and $\\varepsilon_1, \\ldots, \\varepsilon_n \\in \\set{1, -1}$.\nThe *word metric* on $G$ with respect to $S$ is the map $d_S \\colon G \\times G \\to \\Z_{\\ge 0}$ given by\n$$d_S(g, h) = \\abs{g^{-1} h}_S.$$\n\n## Remarks\n\n$d_S$ is a [[metric|0VKD1TP7F3]].\nIt is symmetric because $\\abs{g^{-1}}_S = \\abs{g}_S$, and the triangle inequality holds because $\\abs{gh}_S \\le \\abs{g}_S + \\abs{h}_S$.\nIt is invariant under left multiplication: $d_S(kg, kh) = d_S(g, h)$ for all $k \\in G$.\nIt equals the path distance between $g$ and $h$ in the [[Cayley graph|0GYQK9NDJF]] of $G$ with respect to $S$.\nIf $S$ and $T$ are finite generating sets, there is a constant $C \\ge 1$ with $d_T \\le C d_S$ and $d_S \\le C d_T$.\n\nExample: let $G = \\Z$ and $S = \\set{2, 3}$.\nThen $\\abs{1}_S = 2$: the number $1$ is not in $\\set{\\pm 2, \\pm 3}$, so $\\abs{1}_S \\ge 2$, and $1 = 3 + (-2)$.\nSo $d_S(0, 1) = 2$, while $d_{\\set{1}}(0, 1) = 1$.\n\nNon-example: in $\\Z$, the set $S = \\set{2}$ does not generate $\\Z$.\nThe number $1$ is not a sum of terms $\\pm 2$, so $\\abs{1}_S$ is not defined, and $d_S(0, 1)$ is not a real number.\n",
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