# Cyclic group:Z81

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

This group is defined as the cyclic group of order . It is denoted as , or .

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 81#Arithmetic functions

## Group properties

Properties | Satisfied | Explanation |
---|---|---|

cyclic group | Yes | |

abelian group | Yes | |

metacyclic group | Yes | |

homocyclic group | Yes | |

group of prime power order | Yes | |

nilpotent group | Yes |

## GAP implementation

### Group ID

This finite group has order 81 and has ID 1 among the groups of order 81 in GAP's SmallGroup library. For context, there are 15 groups of order 81. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(81,1)`

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

`gap> G := SmallGroup(81,1);`

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

`IdGroup(G) = [81,1]`

or just do:

`IdGroup(G)`

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

### Other descriptions

This group can be defined using GAP's CyclicGroup function:

`CyclicGroup(81)`