# Cyclic group:Z16

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## Contents

## Definition

The **cyclic group of order sixteen**, denoted or sometimes , is the cyclic group having elements. In other words, it is the quotient of the group of integers by the subgroup of multiples of .

It is given by the presentation:

where is the identity element.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

## Group properties

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

cyclic group | Yes | |

homocyclic group | Yes | |

metacyclic group | Yes | |

abelian group | Yes |

## GAP implementation

### Group ID

This finite group has order 16 and has ID 1 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,1)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(16,1);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [16,1]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Alternative descriptions

Description | Functions used |
---|---|

CyclicGroup(16) |
CyclicGroup |