Definition

A topological group

is profinite if it is isomorphic to the inverse limit of some projective system of finite groups. In other words,

is profinite if there exists a directed set

, 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:

is isomorphic as a group to the projective limit
$\displaystyle \,\underset{\longleftarrow}{\lim}\,H_i := \left\{\left.(h_i) \in ... ...} H_i\ \right\vert\ \alpha_{ij}(h_j) = h_i\ \text{ for all }\ i \leq j\right\}$
under componentwise multiplication.
The isomorphism from to $\,\underset{\longleftarrow}{\lim}\,H_i$ (considered as a subspace of $\prod H_i$ ) is a homeomorphism of topological spaces, where each 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.

(view preamble)

See Also: inverse limit

Other names:

profinite

Also defines:

profinite topology

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

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Cross-references: residually finite, totally disconnected, compact, product topology, discrete topology, topological spaces, homeomorphism, subspace, isomorphism, multiplication, projective limit, group, property, homomorphisms, collection, directed set, words, finite groups, projective system, inverse limit, isomorphic, topological group
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This is version 5 of profinite group, born on 2002-06-26, modified 2007-07-22.
Object id is 3134, canonical name is ProfiniteGroup.
Accessed 7606 times total.

Classification:

AMS MSC:

20E18 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Limits, profinite groups)

Pending Errata and Addenda

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