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  "body": "\n## Summary\n\nA discrete subgroup of a topological group is a subgroup whose elements are isolated from each other.\nIn a Hausdorff topological group, discrete subgroups are closed.\nA discrete subgroup of a Lie group $G$ acts properly discontinuously on each quotient $G/K$ with $K$ compact.\nThe resulting quotient spaces are central objects of geometry and number theory.\nLattices, Fuchsian groups, Kleinian groups and crystallographic groups are all discrete subgroups.\n\n## Explanations\n\n### L1 human\n\nInside the real line, the integers form a subgroup in which each element has some room around it with no other element of the subgroup.\nThat is what discrete means.\nThe numbers $a + b\\sqrt2$ with $a, b$ integers also form a subgroup, but they crowd together everywhere, so this subgroup is not discrete.\nDiscrete subgroups behave like crystals or grids sitting inside a continuous group.\n",
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