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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]] and let $\\Gamma$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThe subgroup $\\Gamma$ is *discrete* if the subspace topology on $\\Gamma$ is the [[discrete topology|0S2TEHX4E1]].\n\n## Remarks\n\nEquivalently, there is an open neighbourhood $U$ of the identity $e$ in $G$ with $U \\cap \\Gamma = \\set{e}$.\nIndeed, then $\\gamma U$ is an open set with $\\gamma U \\cap \\Gamma = \\set{\\gamma}$ for each $\\gamma \\in \\Gamma$.\n\nExample: $\\Z$ is a discrete subgroup of $\\R$, since $U = (-\\tfrac12, \\tfrac12)$ meets $\\Z$ only in $0$.\nLikewise $\\mathrm{SL}_n(\\Z)$ is a discrete subgroup of $\\mathrm{SL}_n(\\R)$, since two distinct integer matrices differ by at least $1$ in some entry.\n\nNon-example: $\\Gamma = \\set{a + b\\sqrt2 : a, b \\in \\Z}$ is a subgroup of $\\R$ that is not discrete.\nThe numbers $(\\sqrt2 - 1)^k$ lie in $\\Gamma$, are nonzero, and tend to $0$, so every neighbourhood of $0$ meets $\\Gamma$ in more than one point.\n\nIn a Hausdorff topological group, every discrete subgroup is closed.\n",
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