---
tag: NR70GHZ37D
kind: problem
name: Hilbert's fifth problem
msc: ["22E05", "22A05", "57S05"]
uses: [92700QAB5Y, 7JBRGCVJRB, QQQ9B1DK4P]
about: [7JBRGCVJRB, 92700QAB5Y]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%27s_fifth_problem
same_as:
  wikidata: Q3893331
  wikipedia: Hilbert's fifth problem
history:
  posed_by: "David Hilbert"
  first_stated: "1900"
---

## Problem

In its standard modern reading, the problem asks the following.
Let $G$ be a [[topological group|92700QAB5Y]] whose underlying space is a [[topological manifold|QQQ9B1DK4P]].
Is $G$ necessarily a [[Lie group|7JBRGCVJRB]]?
That is, does $G$ admit a smooth structure, compatible with its topology, for which multiplication and inversion are smooth?

## Remarks

The answer is yes.
It was obtained in 1952 by Andrew Gleason and by Deane Montgomery and Leo Zippin.
Hidehiko Yamabe later extended their methods to all locally compact groups, giving the [[Gleason–Yamabe theorem|XJF5V9WHKZ]].
A key step is the notion of a group with [[no small subgroups|HPMAJV2RDY]]: a locally compact group with this property is a Lie group.
Hilbert posed the problem in his 1900 list of problems, in terms of continuous transformation groups and without assuming differentiability.
The version for groups acting on manifolds, rather than groups that are manifolds, is the [[Hilbert–Smith conjecture|DS2AS9TSZP]], which is still open.

## Explanations

### L1 human

A Lie group is a group whose elements form a smooth space, like the rotations of the plane or of space.
Smoothness lets one use calculus on the group.
Hilbert 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?
The answer turned out to be yes, so for such groups continuity forces smoothness.