# Direct product of Z4 and Z4

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## Contents

## Definition

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

- It is a homocyclic group of order sixteen and exponent four.
- It is the direct product of two copies of cyclic group:Z4.

## As an abelian group of prime power order

This group is the abelian group of prime power order corresponding to the partition:

In other words, it is the group .

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

generic prime | direct product of cyclic group of prime-square order and cyclic group of prime-square order |

3 | direct product of Z9 and Z9 |

5 | direct product of Z25 and Z25 |

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions

## Group properties

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 16#Group properties

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

Abelian group | Yes | Direct product of cyclic groups | |

Nilpotent group | Yes | Abelian implies nilpotent | |

Metacyclic group | Yes | ||

Homocyclic group | Yes | ||

Supersolvable group | Yes | ||

Solvable group | Yes |

## GAP implementation

### Group ID

This finite group has order 16 and has ID 2 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(16,2)`

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

`gap> G := SmallGroup(16,2);`

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

`IdGroup(G) = [16,2]`

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 also be defined using GAP's DirectProduct function:

DirectProduct(CyclicGroup(4),CyclicGroup(4))