# Cyclic group:Z8

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## Contents

## Definition

The **cyclic group of order eight**, denoted , , or , is defined as the cyclic group of order eight, i.e., it is the quotient of the group of integers by the subgroup of multiples of eight.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#Arithmetic functions

## Group properties

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

cyclic group | Yes | |

abelian group | Yes | |

homocyclic group | Yes | |

metacyclic group | Yes | |

group of prime power order | Yes | |

nilpotent group | Yes |

## GAP implementation

### Group ID

This finite group has order 8 and has ID 1 among the groups of order 8 in GAP's SmallGroup library. For context, there are 5 groups of order 8. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(8,1)`

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

`gap> G := SmallGroup(8,1);`

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

`IdGroup(G) = [8,1]`

or just do:

`IdGroup(G)`

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

### Other descriptions

The group can be described using GAP's CyclicGroup function:

`CyclicGroup(8)`