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  "body": "\n## Statement\n\nLet $G$ be a [[locally compact group|AHV05ZBCYW]].\nThen there is an [[open subgroup|QYCFH68RD1]] $G'$ of $G$ with the following property.\nLet $U$ be any open neighbourhood of the identity in $G'$.\nThen there is a compact [[normal subgroup|0CB0DX5BVV]] $K$ of $G'$ with $K \\subseteq U$ such that $G'/K$ is isomorphic, as a topological group, to a [[Lie group|7JBRGCVJRB]].\n\n## Remarks\n\nThe proof has two parts.\nGleason showed that a locally compact group with [[no small subgroups|HPMAJV2RDY]] is a Lie group, by building one-parameter subgroups and a Lie algebra from the group itself.\nYamabe then showed that every locally compact group can be reduced to this case by dividing out small compact normal subgroups, after passing to an open subgroup.\nExample: for $G = \\Z_p$, the $p$-adic integers, take $G' = G$ and $K = p^n \\Z_p$ inside a given neighbourhood of $0$; then $G/K = \\Z/p^n\\Z$ is a finite group, hence a Lie group of dimension $0$.\nThe open subgroup is needed.\nThe group $G = PSL_2(\\Q_p)$ is simple as an abstract group, so its only compact normal subgroup is trivial.\nBut $G$ is totally disconnected and not discrete, so it is not a Lie group.\nHere one takes for $G'$ a compact open subgroup such as the image of $SL_2(\\Z_p)$, which is profinite.\nA corollary is that a locally compact group with no small subgroups is a Lie group.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that every locally compact group is close to a Lie group.\nAfter passing to an open piece of the group and ignoring an arbitrarily small compact part, what is left is a smooth group.\nSo the smooth groups of calculus and geometry, together with compact groups such as profinite groups, are the building blocks of all locally compact groups.\n\n### L2 human\n\nThe theorem gives the solution of Hilbert's fifth problem: a locally Euclidean group has no small subgroups, so it is a Lie group.\nIt is also the starting point of the structure theory of locally compact groups.\nFor a connected locally compact group one can take $G' = G$, so $G$ is a projective limit of Lie groups.\nThe theorem has found use outside group theory, for example in Breuillard, Green and Tao's classification of approximate groups and in Gromov's theorem on groups of polynomial growth.\n",
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  "name": "Gleason–Yamabe theorem",
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