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  "body": "\n## Summary\n\nThe word metric on a [[group|G5S1M9]] with a generating set $S$ is a distance on the group.\nThe distance from $g$ to $h$ is the least number of generators and their inverses whose product is $g^{-1} h$.\nIt turns the group into a [[metric space|0VKD1TP7F3]], and it equals the path distance in the [[Cayley graph|0GYQK9NDJF]].\nFor a [[finitely generated group|082DWWXMRJ]], changing the finite generating set changes the metric only up to bounded factors.\nSo the metric is well defined up to [[quasi-isometry|6FKYSH729T]].\nIt is the starting point of [[geometric group theory|2T3218RXTD]].\n\n## Explanations\n\n### L1 human\n\nThink of the generators as the allowed steps.\nThe length of an element is the fewest steps needed to reach it from the identity.\nThe distance between two elements is the fewest steps from one to the other.\nIn $\\Z$ with the steps $\\pm 1$, the distance between $m$ and $n$ is $\\abs{m - n}$.\nWith the steps $\\pm 2$ and $\\pm 3$ instead, the number $1$ is at distance $2$ from $0$, since $1 = 3 - 2$ and $1$ itself is not a step.\n",
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