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.

Particular cases

Finite fields

For $q=2$, $PSL(n,q)=SL(n,q)=PGL(n,q)=GL(n,q)$. For $q$ a power of two, $PGL(n,q)=PSL(n,q)=SL(n,q)$ but this is not the same as $GL(n,q)$.

Size of field	Order of matrices	Common name for the projective special linear group
$q$	1	Trivial group
2	2	Symmetric group:S3
3	2	Symmetric group:S4
4	2	Alternating group:A5
5	2	Symmetric group:S5
9	2	Projective general linear group:PGL(2,9)
2	3	Projective special linear group:PSL(3,2)