# Cyclic group:Z9

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## Contents

## Definition

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

## Arithmetic functions

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

order | 9 | |

exponent | 9 | |

derived length | 1 | |

Frattini length | 2 | |

Fitting length | 1 | |

subgroup rank | 1 |

## Group properties

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

cyclic group | Yes | |

elementary abelian group | No | |

abelian group | Yes | |

group of prime power order | Yes | |

nilpotent group | Yes | |

solvable group | Yes |

## GAP implementation

### Group ID

This finite group has order 9 and has ID 1 among the groups of order 9 in GAP's SmallGroup library. For context, there are groups of order 9. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(9,1)`

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

`gap> G := SmallGroup(9,1);`

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

`IdGroup(G) = [9,1]`

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(9)`