# Quasicyclic group

From Groupprops

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 prime number. The **-quasicyclic group** is defined in the following equivalent ways:

- It is the group, under multiplication, of all complex roots of unity for all .
- It is the quotient where is the group of all rational numbers that can be expressed with denominator a power of .
- It is the direct limit of the chain of groups:

.

where the maps are multiplication by maps.

## Particular cases

Prime number | -quasicyclic group |
---|---|

2 | 2-quasicyclic group |

3 | 3-quasicyclic group |

## Group properties

Property | Satisfied? | Explanation | Corollary properties satisfied |
---|---|---|---|

abelian group | Yes | Hence, it is also a nilpotent group and a solvable group. | |

locally cyclic group | Yes | ||

locally finite group | Yes | ||

p-group | Yes | Hence, it is an abelian p-group, so also a nilpotent p-group. |

## Related notions

### Combining quasicyclic groups for all primes

The restricted external direct product of the -quasicyclic groups for all prime numbers is isomorphic to , the group of rational numbers modulo integers.

### p-adics: inverse limit instead of direct limit

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 .