# General linear group:GL(2,Z)

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

The group is defined as the group of invertible matrices over the ring of integers, under matrix multiplication. Since the determinant is multiplicative and the only invertible integers are , this can equivalently be defined as:

.

This is a particular case of a general linear group over integers, which in turn is a particular case of a general linear group.

The subgroup of matrices of determinant one is special linear group:SL(2,Z), and it is a subgroup of index two.

## Arithmetic functions

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

order | Infinite (countable) | |

exponent | Infinite (countable) | |

derived length | not defined | |

Frattini length | not defined | Has a free non-abelian subgroup, so not solvable. |

## Group properties

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

abelian group | No | ||

nilpotent group | No | ||

solvable group | No | ||

perfect group | No |

## GAP implementation

The group can be defined using GAP's GeneralLinearGroup function, as:

`GL(2,Integers)`