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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]] with identity $e$.\n$G$ is *torsion-free* if every element $g \\ne e$ of $G$ has infinite [[order|0WE2NHZJ7B]], that is, $g^n \\ne e$ for all integers $n \\ge 1$.\n\n## Remarks\n\nExample: the [[integers|0C4JS9RF5M]] $\\Z$ under addition.\nIf $m \\ne 0$ and $n \\ge 1$, then $n m \\ne 0$.\nThe groups $\\Z^k$, the [[rational numbers|0NCJ5WXN6Z]] under addition, and [[free groups|0KTW26C80J]] are also torsion-free.\n\nNon-example: $\\Z \\times \\Z/2\\Z$.\nThe element $(0, 1)$ is not the identity, but $2 \\cdot (0, 1) = (0, 0)$.\n\nFor an abelian group $A$ with [[torsion subgroup|0DBWXQRXQJ]] $T$, the quotient $A/T$ is torsion-free.\nBy the [[structure theorem for finitely generated abelian groups|0RY0B72XAS]], a finitely generated torsion-free abelian group is isomorphic to $\\Z^n$ for some $n \\ge 0$.\nIn a nonabelian group the elements of finite order need not form a subgroup.\nIn the infinite dihedral group, two reflections have order $2$, but their product is a translation of infinite order.\n",
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