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A pro-p group is a profinite group that is an inverse limit of finite p-groups
Definition
A pro- group is a topological group that is topologically isomorphic to an inverse limit of finite -groups. In particular every pro- group is a profinite group in the sense of A profinite group is a topological group isomorphic to an inverse limit of finite discrete groups.
The prime is part of the structure: a group is pro- only when the finite quotients in the chosen inverse-limit presentation are all -groups.
Depends on
Used by
- The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod pⁿ Definition
- The pro-p completion of an abstract group is the inverse limit over its finite p-group quotients Definition
- A finite p-group is naturally isomorphic to its own pro-p completion Example
- Every profinite group is pro-p for some prime False statement
- A topological group is pro-p exactly when it is profinite and has an open normal basis with finite p-group quotients Theorem
Dependency tree · two levels
3 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
- Gareth Wilkes, Profinite Groups and Group Cohomology lecture notes (standard reference, not scraped)
- Brian Osserman, Inverse limits and profinite groups (standard reference, not scraped)