Topological groups, Lie groups Topological groups: basic notions Example

Cantor group

Builds on Product topology Cantor set Haar measure No small subgroups

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.

Check that it is a profinite group. 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, via $(a_k) \mapsto \sum_k 2 a_k 3^{-k-1}$ . The normalized Haar measure 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 condition and is far from a Lie group.

Reading path

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Cantor set example
Haar measure concept
No small subgroups concept
Product topology concept
Cantor group example

Tag	`60G5ABJQ75`
URI	`m2://60G5ABJQ75`
Kind	example
Writing	W0 stub
MSC	22C05, 20E18, 54H11
Instance of	Profinite group
Uses	Product topology, Cantor set, Haar measure, No small subgroups
Changed	2 November 2026

Cantor group

Example

Remarks

Reading path

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