# General linear group of degree two

From Groupprops

## Contents

## 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.

## Particular cases

### Finite fields

Size of field | Common name for general linear group of degree two |
---|---|

symmetric group:S3 | |

general linear group:GL(2,3) | |

general linear group:GL(2,4) | |

general linear group:GL(2,5) |

### Infinite rings and fields

Name of ring/field | Common name for general linear group of degree two |
---|---|

Ring of integers | general linear group:GL(2,Z) |

Field of rational numbers | general linear group:GL(2,Q) |

Field of real numbers | general linear group:GL(2,R) |

Field of complex numbers | general linear group:GL(2,C) |

## Arithmetic functions

Here, denotes the order of the finite field and the group we work with is . is the characteristic of the field, i.e., it is the prime whose power 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 . All elements have order dividing or . | |

number of conjugacy classes | There are conjugacy classes of semisimple matrices and conjugacy classes of matrices with repeated eigenvalues. |

## 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. |

## Elements

`Further information: Element structure of general linear groups of degree two`

The elements are as follows:

- There are conjugacy classes of size one, corresponding to the central elements.
- There are conjugacy classes of size each. These are obtained as follows: we know that there is a field extension of size , which can be identified with a vector space of dimension two over the field of elements. Left multiplication by an element in the field gives a matrix. There are of these elements corresponding to a particular choice of basis for the field that are
*not*scalar matrices. These come in pairs of conjugate elements (conjugate in the sense of field extensions) that are hence also conjugate in . Thus, there are conjugacy classes here. Further, for each such element, the centralizer is the multiplicative group of the field with elements, which has order . The quotient has order . - There are conjugacy classes of size each. These correspond to elements that are diagonalizable with distinct eigenvalues. The number corresponds to choices of two distinct elements among the . Further, the centralizer of a diagonal matrix is the group of diagonal matrices, and has order . The quotient has order .
- There are conjugacy classes of size each.

## Subgroup-defining functions

Subgroup-defining function | Value | Explanation |
---|---|---|

Center | The subgroup of scalar matrices. Cyclic of order | Center of general linear group is group of scalar matrices over center. |

Commutator subgroup | Except the case of , it is the special linear group of degree two, which has index . | Commutator subgroup of general linear group is special linear group |

## Quotient-defining functions

Subgroup-defining function | Value | Explanation |
---|---|---|

Inner automorphism group | Projective general linear group of degree two | Quotient by the center, which is the group of scalar matrices. |

Abelianization | This is isomorphic to the multiplicative group of the field. | Quotient by the commutator subgroup, which is the special linear group, which is the kernel of the determinant map that surjects to the multiplicative group of the field. |