# Cyclic group:Z5

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## Contents

## Definition

This group, denoted , is defined in the following equivalent ways:

- It is the cyclic group of order five.
- It is the additive group of the field of five elements.

## Arithmetic functions

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

order | 5 | |

exponent | 5 | |

derived length | 1 | The group is abelian. |

Frattini length | 1 | |

Fitting length | 1 |

## Group properties

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

cyclic group | Yes | |

abelian group | Yes | |

nilpotent group | Yes | |

solvable group | Yes |

## GAP implementation

### Group ID

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

`SmallGroup(5,1)`

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

`gap> G := SmallGroup(5,1);`

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

`IdGroup(G) = [5,1]`

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 constructed using GAP's CyclicGroup function:

`CyclicGroup(5)`