---
tag: HPMAJV2RDY
kind: concept
name: No small subgroups
msc: ["22A05", "22E05", "22D05"]
sources:
  - url: https://en.wikipedia.org/wiki/No_small_subgroup
same_as:
  wikidata: Q7045262
  wikipedia: No small subgroup
---

## Summary

A [[topological group|92700QAB5Y]] has no small subgroups if some neighbourhood of the identity contains no subgroup other than the trivial one.
Every [[Lie group|7JBRGCVJRB]] has this property.
For locally compact groups the converse holds as well, by the work of Gleason and Yamabe.
This is the key to the solution of Hilbert's fifth problem.

## Explanations

### L1 human

In the real line, the interval $(-1, 1)$ contains no subgroup except $\set{0}$: any subgroup containing a number $x \ne 0$ also contains $x, 2x, 3x, \dots$, which eventually leave the interval.
Groups built from smooth spaces all behave this way near the identity.
In the $p$-adic integers, by contrast, every small ball around $0$ is itself a subgroup.
So "no small subgroups" is a simple test that separates smooth groups from groups of this other kind.