# General linear group of degree two

From Groupprops

## Definition

The **general linear group of degree two** over a field (respectively, over a unital ring ), is defined as the group, under multiplication, of invertible matrices with entries in . It is denoted (respectively, ).

For a prime power , or denotes the general linear group of degree two over the field (unique up to isomorphism) with elements.

## Arithmetic functions

Here, denotes the order of the finite field and the group we work with is .

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

order | options for first row, options for second row. | |

exponent | There is an element of order and an element of order . | |

number of conjugacy classes | There are conjugacy classes of semisimple matrices and conjugacy classes of unipotent matrices. |

## Group properties

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

Abelian group | No | The matrices and don't commute. |

Nilpotent group | No | is simple for , and we can check the cases separately. |

Solvable group | Yes if , no otherwise. | is simple for . |

Supersolvable group | Yes if , no otherwise. | is simple for , and we can check the cases separately. |