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  "body": "\n## Problem\n\nIn its standard modern reading, the problem asks the following.\nLet $G$ be a [[topological group|92700QAB5Y]] whose underlying space is a [[topological manifold|QQQ9B1DK4P]].\nIs $G$ necessarily a [[Lie group|7JBRGCVJRB]]?\nThat is, does $G$ admit a smooth structure, compatible with its topology, for which multiplication and inversion are smooth?\n\n## Remarks\n\nThe answer is yes.\nIt was obtained in 1952 by Andrew Gleason and by Deane Montgomery and Leo Zippin.\nHidehiko Yamabe later extended their methods to all locally compact groups, giving the [[Gleason–Yamabe theorem|XJF5V9WHKZ]].\nA key step is the notion of a group with [[no small subgroups|HPMAJV2RDY]]: a locally compact group with this property is a Lie group.\nHilbert posed the problem in his 1900 list of problems, in terms of continuous transformation groups and without assuming differentiability.\nThe version for groups acting on manifolds, rather than groups that are manifolds, is the [[Hilbert–Smith conjecture|DS2AS9TSZP]], which is still open.\n\n## Explanations\n\n### L1 human\n\nA Lie group is a group whose elements form a smooth space, like the rotations of the plane or of space.\nSmoothness lets one use calculus on the group.\nHilbert asked whether smoothness is automatic: if a group is only known to look locally like ordinary space, with continuous operations, must it already be smooth?\nThe answer turned out to be yes, so for such groups continuity forces smoothness.\n",
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