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  "body": "\n## Summary\n\nA [[topological group|92700QAB5Y]] has no small subgroups if some neighbourhood of the identity contains no subgroup other than the trivial one.\nEvery [[Lie group|7JBRGCVJRB]] has this property.\nFor locally compact groups the converse holds as well, by the work of Gleason and Yamabe.\nThis is the key to the solution of Hilbert's fifth problem.\n\n## Explanations\n\n### L1 human\n\nIn the real line, the interval $(-1, 1)$ contains no subgroup except $\\set{0}$: any subgroup containing a number $x \\ne 0$ also contains $x, 2x, 3x, \\dots$, which eventually leave the interval.\nGroups built from smooth spaces all behave this way near the identity.\nIn the $p$-adic integers, by contrast, every small ball around $0$ is itself a subgroup.\nSo \"no small subgroups\" is a simple test that separates smooth groups from groups of this other kind.\n",
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