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 $P G L (n, k)$ 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 $G L (n, k)$ 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 $P G L (n, q)$ the group $P G L (n, F_{q})$ where $F_{q}$ 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 $P G L (V)$ , is defined as the inner automorphism group of $G L (V)$ , viz the quotient of $G L (V)$ by its center, which is the group of scalar multiples of the identity transformation.

Particular cases

Particular cases by degree

Degree $n$	Information on projective general linear group $P G L (n, k)$ 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

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 $P G L (n, q) = P G L (n, F_{q})$	Order of $P G L (n, q)$
$q$	$p$	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