# Projective general linear group

*This term associates to every field, a corresponding group property. In other words, given a field, every group either has the property with respect to that field or does not have the property with respect to that field*

*This group property is natural number-parametrized, in other words, for every natural number, we get a corresponding group property*

## Contents

## Definition

### In terms of dimension

Let be a natural number and be a field. The **projective general linear group** of order over , denoted is defined in the following equivalent ways:

- It is the group of automorphisms of projective space of dimension , that arise from linear automorphisms of the vector space of dimension .
- It is the quotient of by its center, viz the group of scalar multiplies of the identity (isomorphic to the group )

For a prime power, we denote by the group where is the field (unique up to isomorphism) of size .

### In terms of vector spaces

Let be a vector space over a field . The projective general linear group of , denoted , is defined as the inner automorphism group of , viz the quotient of by its center, which is the group of scalar multiples of the identity transformation.

## Arithmetic functions

### Over a finite field

Below is information for the projective general linear group of degree over a finite field of size .

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

order | has the same order | The order of is . The center has order , so the quotient has order equal to the quotient of the order of by . | |

number of conjugacy classes | (no good generic expression, but overall it's a polynomial in of degree ) | See element structure of projective general linear group over a finite field |

## Particular cases

### Particular cases by degree

Degree | Information on projective general linear group over a field |
---|---|

1 | trivial group always |

2 | projective general linear group of degree two |

3 | projective general linear group of degree three |

4 | projective general linear group of degree four |

### Finite fields

If , then for all .

More generally, if is relatively prime to , then the groups are all isomorphic to each other. However, they are not isomorphic to .

Size of field | Characteristic of field (so is a power of | Degree of projective general linear group | Common name for the projective general linear group | Order of |
---|---|---|---|---|

1 | Trivial group | 1 | ||

2 | 2 | 2 | Symmetric group:S3 | 6 |

3 | 3 | 2 | Symmetric group:S4 | 24 |

4 | 2 | 2 | Alternating group:A5 | 60 |

5 | 5 | 2 | Symmetric group:S5 | 120 |

9 | 3 | 2 | Projective general linear group:PGL(2,9) | 720 |

2 | 2 | 3 | Projective special linear group:PSL(3,2) | 168 |