Projective general linear group

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.

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

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 $$GL(n,q)$$.