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  "body": "\n## Summary\n\nA maximal compact subgroup of a [[topological group|92700QAB5Y]] is a compact subgroup that is not contained in any larger compact subgroup.\nIn a connected [[Lie group|7JBRGCVJRB]], maximal compact subgroups exist and are all conjugate.\nThe group is then, as a space, the product of a maximal compact subgroup with a Euclidean space.\nSo a maximal compact subgroup carries the topology of the whole group.\n\n## Explanations\n\n### L1 human\n\nInside a big group of symmetries, look for the largest piece that is bounded.\nFor the invertible real $n \\times n$ matrices, this piece is the group $\\mathrm{O}(n)$ of rotations and reflections.\nEvery invertible matrix is a rotation or reflection times a stretching along perpendicular axes.\nThe stretchings form a space with no holes, so all the interesting shape of the group comes from $\\mathrm{O}(n)$.\n",
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