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