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  "body": "\n## Definition\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$.\nA *Zassenhaus neighbourhood* of $G$ is an open [[neighbourhood|0ZCY0Z80B6]] $U$ of $e$ with the following property.\nFor every [[discrete subgroup|5M7D1BSHVZ]] $\\Gamma$ of $G$, the subgroup generated by $\\Gamma \\cap U$ is [[nilpotent|0Q7MMQHC1P]].\n\n## Remarks\n\nEvery Lie group has a Zassenhaus neighbourhood, by a theorem of Zassenhaus.\nOne can moreover choose $U$ so that $\\Gamma \\cap U$ lies in a connected nilpotent Lie subgroup of $G$.\n\nExample: in $G = \\R^n$, every open neighbourhood of $0$ is a Zassenhaus neighbourhood.\nIndeed every subgroup of $\\R^n$ is abelian, hence nilpotent.\n\nNon-example: $U = \\mathrm{SO}(3)$ in $G = \\mathrm{SO}(3)$.\nThe rotation group of the icosahedron is a finite, hence discrete, subgroup isomorphic to the alternating group $A_5$.\nIt lies in $U$ and is not nilpotent, since $A_5$ is simple and non-abelian.\n",
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