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A profinite group is a topological group isomorphic to an inverse limit of finite discrete groups
Definition
A profinite group is a topological group that is topologically isomorphic to an inverse limit of finite discrete groups.
Because inverse limits on this page are given concretely as compatible tuples, this means precisely that the group admits a homeomorphic group isomorphism onto some as in Inverse limits of finite discrete groups are Hausdorff and totally disconnected, and compact assuming Choice and Topological group: multiplication and inversion are continuous.
Depends on
Used by
- The profinite completion is the inverse limit of the finite quotients G over N Definition
- The kernels of the finite coordinate projections form an open normal neighbourhood basis at the identity Lemma
- Assuming Choice, a topological group is profinite exactly when it is compact, Hausdorff, and totally disconnected Theorem
- The profinite completion is initial among continuous homomorphisms from G to profinite groups Theorem
Dependency tree · two levels
13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
- H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)