Projective general linear group of degree two

Definition
For a field $$k$$, the projective general linear group of degree two $$PGL(2,k)$$ or $$PGL_2(k)$$ is defined as the quotient group of the general linear group of degree two $$GL(2,k)$$ by its center, which is the group of scalar matrices in it (because center of general linear group is group of scalar matrices over center).

In other words, it is the inner automorphism group of the general linear group of degree two.

The projective general linear group of degree two is the degree two special case of the projective general linear group.

For $$q$$ a prime power, the projective general linear group of degree two denoted $$PGL(2,q)$$ is defined as the projective general linear group of degree two over the field $$\mathbb{F}_q$$ of $$q$$ elements (unique up to field isomorphism).

Over a finite field
Here, $$q$$ is the size of the finite field for which we consider the group $$PGL(2,q)$$. $$p$$ is the characteristic of the field, so $$q$$ is a power of $$p$$.

Finite fields
Note that for $$q = 2$$, $$PGL(2,q)$$ is isomorphic to all of $$GL(2,q), SL(2,q), PSL(2,q)$$. For $$q$$ a power of 2, $$PGL(2,q)$$ is isomorphic to $$PSL(2,q)$$ and $$SL(2,q)$$ but not to $$GL(2,q)$$.

For $$q$$ not a power of $$2$$, $$PGL(2,q)$$ admits $$PSL(2,q)$$ as a subgroup of index two. $$SL(2,q)$$ and $$PGL(2,q)$$ have the same order, and in fact the same composition factors, but are non-isomorphic, because $$SL(2,q)$$ admits $$PSL(2,q)$$ as a quotient rather than as a subgroup. In general, for $$q \ge 5$$, $$PGL(2,q)$$ is an almost simple group and $$SL(2,q)$$ is a quasisimple group.

Over a finite field
Below is a summary of the linear representation theory of $$PGL(2,q)$$: