# 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

## Contents

## Definition

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

This group is the additive group of the ring of Witt vectors over the prime field .

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order of a profinite group | for the relevant prime . | This is a supernatural number as is the convention for orders of profinite groups. |

## Powering

The additive group of -adic integers is powered over the set of all primes other than .

## Group properties

### Abstract group properties

Property | Satisfied? | Explanation |
---|---|---|

abelian group | Yes | |

aperiodic group | Yes | No nonzero element has finite order. |

### Topological group properties

Here, the topology is from the profinite group structure.

Property | Satisfied? | Explanation |
---|---|---|

profinite group | Yes | By definition |

pro-p-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. |

## Related notions

### 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.

### Dual concept

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 .