Additive group of p-adic integers
This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups
- 1 Definition
- 2 Arithmetic functions
- 3 Powering
- 4 Group properties
- 5 Related notions
Let be a (fixed here) prime number. This is a group determined uniquely up to isomorphism based on and is sometimes denoted (though that notation is also used for other things, and we can best infer meaning from context).
As an inverse limit
The additive group of p-adic integers is a profinite group defined as the inverse limit of the inverse system:
where each of the maps:
reduces an integer mod to its value mod .
Note that this definition also endows the group with a topology as a profinite group. In this topology, two elements are close if they agree mod for large .
As sequences with cumulative information
The additive group of p-adic integers is the set of sequences:
where is an integer mod , the addition is coordinate-wise (with each coordinate addition in the integers mod ), and for , reducing mod yields .
As sequences with carries
The additive group of p-adic integers is the set of formal sums:
where , and the addition is done with carries, i.e., to add two sequences and , we add coordinate-wise and if any of the sums is or more, we take a carry of 1 to the next sum.
As the additive group of the ring of Witt vectors
|order of a profinite group||for the relevant prime .||This is a supernatural number as is the convention for orders of profinite groups.|
The additive group of -adic integers is powered over the set of all primes other than .
Abstract group properties
|aperiodic group||Yes||No nonzero element has finite order.|
Topological group properties
Here, the topology is from the profinite group structure.
|profinite group||Yes||By definition|
|compact group||Yes||profinite groups are compact|
|T0 topological group||Yes||profinite groups are Hausdorff, hence|
|totally disconnected group||Yes||profinite groups are totally disconnected|
|topologically cyclic group||Yes||, the subgroup generated by the element that projects to 1 mod for all , is a dense subgroup. In fact, this group can be thought of as a compactification of .|
|topologically finitely generated group||Yes||Follows from being topologically cyclic.|
|connected topological group||No||It is nontrivial and totally disconnected.|
Combining all primes
The profinite completion of the integers, denoted is a group obtained by taking the profinite completion of the group of integers, which is a residually finite group. Intuitively, this group is obtained by "completing" the group of integers simultaneously at all primes. It turns out that the profinite completion of the integers can be viewed as the external direct product, over all primes , of the additive group of -adic integers.
The additive group of p-adic integers can, in a vague sense, be considered to be constructed using a method dual to the method used to the quasicyclic group. While the -adics are constructed as an inverse limit for surjective maps , the quasicyclic group is constructed as a direct limit for injective maps .