# 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

## Definition

### In terms of dimension

Let $n$ be a natural number and $k$ be a field. The projective general linear group of order $n$ over $k$, denoted $PGL(n,k)$ is defined in the following equivalent ways:

• It is the group of automorphisms of projective space of dimension $n-1$, that arise from linear automorphisms of the vector space of dimension $n$.
• It is the quotient of $GL(n,k)$ by its center, viz the group of scalar multiplies of the identity (isomorphic to the group $k^*$)

For $q$ a prime power, we denote by $PGL(n,q)$ the group $PGL(n,\mathbb{F}_q)$ where $\mathbb{F}_q$ is the field (unique up to isomorphism) of size $q$.

### In terms of vector spaces

Let $V$ be a vector space over a field $k$. The projective general linear group of $V$, denoted $PGL(V)$, is defined as the inner automorphism group of $GL(V)$, viz the quotient of $GL(V)$ 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 $n$ over a finite field of size $q$.

Function Value Similar groups Explanation
order $q^{\binom{n}{2}}\prod_{i=2}^n (q^i - 1)$ $SL(n,q)$ has the same order The order of $GL(n,q)$ is $q^{\binom{n}{2}}\prod_{i=1}^n (q^i - 1)$. The center has order $q - 1$, so the quotient has order equal to the quotient of the order of $GL(n,q)$ by $q - 1$.
number of conjugacy classes (no good generic expression, but overall it's a polynomial in $q$ of degree $n - 1$) See element structure of projective general linear group over a finite field

## Particular cases

### Particular cases by degree

Degree $n$ Information on projective general linear group $PGL(n,k)$ over a field $k$
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 $q = 2$, then $PGL(n,q) = GL(n,q) = SL(n,q) = PSL(n,q)$ for all $n$.

More generally, if $q - 1$ is relatively prime to $n$, then the groups $PGL(n,q), SL(n,q), PSL(n,q)$ are all isomorphic to each other. However, they are not isomorphic to $GL(n,q)$.

Size of field $q$ Characteristic $p$ of field (so $q$ is a power of $p$ Degree of projective general linear group $n$ Common name for the projective general linear group $PGL(n,q) = PGL(n,\mathbb{F}_q)$ Order of $PGL(n,q)$ $q$ $p$ 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