# Cyclic group of prime-square order

From Groupprops

## Contents

## Definition

Let be a prime number. The **cyclic group** of order , denoted , is defined as the cyclic group with elements.

This is the cyclic group corresponding to the partition of .

## Particular cases

prime number | corresponding cyclic group of prime-square order |
---|---|

2 | cyclic group:Z4 |

3 | cyclic group:Z9 |

5 | cyclic group:Z25 |

7 | cyclic group:Z49 |

## Arithmetic functions

## GAP implementation

### Group ID

This finite group has order p^2 and has ID 1 among the group of order p^2 in GAP's SmallGroup library. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(p^2,1)`

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

`gap> G := SmallGroup(p^2,1);`

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

`IdGroup(G) = [p^2,1]`

or just do:

`IdGroup(G)`

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

### Short descriptions

Description | Functions used | Mathematical comments |
---|---|---|

CyclicGroup(p^2) |
CyclicGroup |