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  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]].\n$G$ is a *torsion group* (or *periodic group*) if every element of $G$ has finite [[order|0WE2NHZJ7B]].\nThat is, for every $g \\in G$ there is an integer $n \\ge 1$ with $g^n = e$.\n\n## Remarks\n\nExample: the group $\\Q/\\Z$, the [[rational numbers|0NCJ5WXN6Z]] modulo the [[integers|0C4JS9RF5M]] under addition.\nIf $x = \\tfrac{a}{b} + \\Z$ with $a, b \\in \\Z$ and $b \\ge 1$, then $b x = a + \\Z = 0$, so $x$ has finite order.\nThe Prüfer group is another infinite example.\n\nNon-example: the group $\\Z$ under addition.\nFor $n \\ne 0$ and $k \\ge 1$, $k n \\ne 0$, so $n$ has infinite order.\n\nA group of finite [[exponent|0ZDW0H0VX4]] is a torsion group, but $\\Q/\\Z$ is a torsion group of infinite exponent.\nA finitely generated abelian torsion group is finite.\nFor nonabelian groups this fails: there are finitely generated infinite torsion groups, such as the Grigorchuk group; see the Burnside problem.\n",
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