# Group of integers

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## Contents

## Definition

### Usual definition

The **group of integers**, typically denoted , is defined as follows:

- The underlying set is the set of all integers
- The group operation is integer addition
- The identity element is the integer
- The inverse map is the additive inverse, sending an integer to the integer

In the 4-tuple notation, the group of integers in the group .

### Other definitions

Some other equivalent formulations of the group of integers:

- It is the additive group of the ring of integers
- It is the infinite cyclic group
- It is the free group on one generator
- It is the free abelian group on one generator

## Arithmetic functions

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

order ((number of elements, equivalently, cardinality or size of underlying set) | Infinite (countable) | Not a finite group. |

exponent | Infinite | Not a periodic group. |

derived length | 1 | The group is an abelian group. |

nilpotency class | 1 | The group is an abelian group. |

Fitting length | 1 | The group is an abelian group. |

Frattini length | 1 | The group is a Frattini-free group. |

subgroup rank of a group | 1 | The group is cyclic, hence so is every subgroup. |

## Group properties

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

cyclic group | Yes | ||

abelian group | Yes | Cyclic implies abelian | |

finite group | No | ||

finitely generated group | Yes | Generating set of size one. | |

slender group | Yes | Every subgroup is cyclic. | |

Hopfian group | Yes | Not isomorphic to any proper quotient, which is finite. | |

co-Hopfian group | No | Isomorphic to the proper subgroup generated by any element not the generator or the identity. |

## GAP implementation

The group can be defined using the FreeGroup function:

`FreeGroup(1)`