---
tag: 60G5ABJQ75
kind: example
name: Cantor group
msc: ["22C05", "20E18", "54H11"]
instance_of: [KA0YDB6QXE]
uses: [0GBG8B49AA, 0BXNDZFQME, VY0HXD3GC4, HPMAJV2RDY]
sources:
  - url: https://en.wikipedia.org/wiki/Cantor_space
---

## Example

The *Cantor group* is the product $C = (\Z/2\Z)^{\N}$ of countably many copies of the group of order $2$.
Its elements are sequences $(a_0, a_1, a_2, \dots)$ with $a_k \in \set{0, 1}$, added coordinatewise modulo $2$.
Each factor has the discrete topology, and $C$ has the [[product topology|0GBG8B49AA]].

Check that it is a [[profinite group|KA0YDB6QXE]].
Addition and negation are continuous, since in each coordinate they depend on only one coordinate of the inputs.
The space $C$ is compact by Tychonoff's theorem and Hausdorff as a product of Hausdorff spaces.
It is totally disconnected: two different sequences differ in some coordinate $k$.
The sets where $a_k = 0$ and where $a_k = 1$ are open, closed and separate them.
A compact Hausdorff totally disconnected group is profinite.
Equivalently, $C$ is the inverse limit of the finite groups $(\Z/2\Z)^n$ under the maps that forget the last coordinate.

## Remarks

As a topological space, $C$ is homeomorphic to the [[Cantor set|0BXNDZFQME]], via $(a_k) \mapsto \sum_k 2 a_k 3^{-k-1}$.
The normalized [[Haar measure|VY0HXD3GC4]] of $C$ is the product of the uniform measures on $\set{0, 1}$, the law of an infinite sequence of fair coin tosses.
Every neighbourhood of $0$ contains a subgroup of the form $\set{a : a_0 = \cdots = a_n = 0}$, so $C$ fails the [[no small subgroups|HPMAJV2RDY]] condition and is far from a Lie group.

## Explanations

### L1 human

Think of an infinite sequence of light switches, each on or off.
Adding two such sequences means flipping the switches that are on in the second one.
Two sequences are close when they agree on a long first stretch of switches.
This gives a group whose space of points is the Cantor set, the classical dust of points obtained by removing middle thirds.