Definition

A topological group $G$ is profinite if it is isomorphic to the inverse limit of some projective system of finite groups. In other words, $G$ is profinite if there exists a directed set $I$ , a collection of finite groups $\{H_i\}_{i \in I}$ , and homomorphisms $\alpha_{ij}\colon H_j \to H_i$ for each pair $i,j \in I$ with $i \leq j$ , satisfying

$\alpha_{ii} = 1$ for all $i \in I$ ,
$\alpha_{ij} \circ \alpha_{jk} = \alpha_{ik}$ for all $i,j,k \in I$ with $i \leq j \leq k$ ,

with the property that:

$G$ is isomorphic as a group to the projective limit $$ \ilim H_i := \left\{\left.(h_i) \in \prod_{i \in I} H_i\ \right|\ \alpha_{ij}(h_j) = h_i\ \text{ for all }\ i \leq j\right\} $$ under componentwise multiplication.
The isomorphism from $G$ to $\ilim H_i$ (considered as a subspace of $\prod H_i$ ) is a homeomorphism of topological spaces, where each $H_i$ is given the discrete topology and $\prod H_i$ is given the product topology.

The topology on a profinite group is called the profinite topology.

Properties

One can show that a topological group is profinite if and only if it is compact and totally disconnected. Moreover, every profinite group is residually finite.

"profinite group" is owned by djao.

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Other names:

profinite

Also defines:

profinite topology

Attachments:: pro- group (Definition) by alozano
procyclic group (Definition) by alozano

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AMS MSC:	20E18 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Limits, profinite groups)
	22C05 (Topological groups, Lie groups :: Compact groups)

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