# Cyclic group:Z32

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## Contents

## Definition

This group, denoted , , or , is the cyclic group of order .

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions

## Group properties

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

cyclic group | Yes | |

abelian group | Yes | |

homocyclic group | Yes | |

metacyclic group | Yes | |

directly indecomposable group | Yes | |

centrally indecomposable group | Yes | |

splitting-simple group | Yes |

## GAP implementation

### Group ID

This finite group has order 32 and has ID 1 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(32,1)`

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

`gap> G := SmallGroup(32,1);`

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

`IdGroup(G) = [32,1]`

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 described using GAP's CyclicGroup function:

`CyclicGroup(32)`