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  "body": "\n## Summary\n\nA compact group is a topological group whose topology is compact and Hausdorff.\nCompact groups carry a Haar measure of total mass $1$, so one can average over the group as over a finite group.\nSo their representation theory looks much like that of finite groups.\nEvery finite-dimensional representation splits into irreducible ones, and the Peter–Weyl theorem decomposes $L^2(G)$.\n\n## Explanations\n\n### L1 human\n\nA compact group is a group that is also a \"closed and bounded\" space, with no way to run off to infinity.\nThe circle of rotations of the plane is the simplest infinite example, and the group of rotations of space is another.\nEvery finite group is a compact group with the discrete topology.\nThe key feature is that one can take the average of a function over all elements of the group, just as one averages over a finite set.\n",
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