# Direct product of cyclic group of prime-square order and cyclic group of prime order

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.

View other such prime-parametrized groups

## Contents

## Definition

Let be a prime number. This group of order is defined as the external direct product of the cyclic group of prime-square order and the cyclic group of prime order.

## Particular cases

Value of prime number | Corresponding group |
---|---|

2 | direct product of Z4 and Z2 |

3 | direct product of Z9 and Z3 |

5 | direct product of Z25 and Z5 |

## Arithmetic functions

Compare and contrast arithmetic function values with other groups of prime-cube order at Groups of prime-cube order#Arithmetic functions

### Arithmetic functions taking values between 0 and 3

### Arithmetic functions of a counting nature

Note that since the group is abelian, the number of subgroups equals the number of conjugacy classes of subgroups as well as the number of normal subgroups.

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

number of subgroups | ||

number of automorphism classes of subgroups | 6 | |

number of characteristic subgroups | 4 |

## GAP implementation

### Group ID

This finite group has order p^3 and has ID 2 among the group of order p^3 in GAP's SmallGroup library. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(p^3,2)`

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

`gap> G := SmallGroup(p^3,2);`

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

`IdGroup(G) = [p^3,2]`

or just do:

`IdGroup(G)`

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