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  "body": "\n## Definition\n\nLet $S$ be a set.\nThe free group $F_S$ on $S$ is the set of [[reduced words|0KBTP8Q2FS]] on $S$.\nThe product of two reduced words is their concatenation, followed by reduction.\nA [[group|G5S1M9]] $G$ is free when it is [[isomorphic|08MG7H8JW5]] to $F_S$ for some set $S$.\n\n## Remarks\n\nThe identity of $F_S$ is the empty word, and the inverse of $t_1 \\cdots t_n$ is $t_n^{-1} \\cdots t_1^{-1}$.\nThe set $S$ is a basis of $F_S$. Also $F_S \\cong F_T$ if and only if $S$ and $T$ have the same cardinality, which is the rank.\nThe free group of rank $1$ is the [[infinite cyclic group|02B1T50JD1]], and for $|S| \\geq 2$ the centre of $F_S$ is trivial.\nA nontrivial finite group is never free, because each element of a basis has infinite order.\n\n## Explanations\n\n### L1 human\n\nA free group has generators but no relations that you do not need.\nTwo words give the same element only when the group axioms force it, as in $s t = s u u^{-1} t$.\nEach group is a quotient of a free group, so free groups are the starting point for presentations.\n",
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