# Direct product of Z10 and Z2

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## Contents

## Definition

This group is defined in the following equivalent ways:

- It is the direct product of the cyclic group of order ten and the cyclic group of order two.
- It is the direct product of the cyclic group of order five and the Klein four-group.
- It is the direct product of the cyclic group of order five and two copies of the cyclic group of order two.

## Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order (number of elements, equivalently, cardinality or size of underlying set) | 20 | groups with same order | |

exponent of a group | 10 | groups with same order and exponent of a group | groups with same exponent of a group | |

Frattini length | 1 | groups with same order and Frattini length | groups with same Frattini length | |

Fitting length | 1 | groups with same order and Fitting length | groups with same Fitting length | |

derived length | 1 | groups with same order and derived length | groups with same derived length | |

minimum size of generating set | 2 | groups with same order and minimum size of generating set | groups with same minimum size of generating set |

## Group properties

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

cyclic group | No | |

abelian group | Yes | |

nilpotent group | Yes |

## GAP implementation

### Group ID

This finite group has order 20 and has ID 5 among the groups of order 20 in GAP's SmallGroup library. For context, there are 5 groups of order 20. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(20,5)`

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

`gap> G := SmallGroup(20,5);`

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

`IdGroup(G) = [20,5]`

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 described using GAP's DirectProduct, CyclicGroup and ElementaryAbelianGroup functions, in any of these ways:

`DirectProduct(CyclicGroup(10),CyclicGroup(2))`

`DirectProduct(CyclicGroup(5),ElementaryAbelianGroup(4))`

`DirectProduct(CyclicGroup(5),CyclicGroup(2),CyclicGroup(2))`