Projective special linear group of degree two

For a field or commutative unital ring
The projective special linear group of degree two over a field $$k$$, or more generally over a commutative unital ring $$R$$, is defined as the quotient of the defining ingredient::special linear group of degree two over the same field or commutative unital ring by the subgroup of scalar matrices in that group. The group is denoted by $$PSL(2,R)$$ or $$PSL_2(R)$$.

For a prime power
Suppose $$q$$ is a prime power. The projective special linear group $$PSL(2,q)$$ is defined as the projective special linear group of degree two over the field (unique up to isomorphism) with $$q$$ elements.

For prime powers $$q$$
Note that for $$q = 2$$, $$PSL(2,q) \cong PGL(2,q) \cong SL(2,q) \cong GL(2,q)$$. Also, for $$q$$ a power of 2 (so $$p = 2$$), $$PSL(2,q) \cong PGL(2,q) \cong SL(2,q)$$ and $$GL(2,q) \cong PSL(2,q) \times \mathbb{F}_q^\ast$$.

Arithmetic functions
Below we give the arithmetic functions for $$PSL(2,q)$$, where $$q$$ is a power $$p^r$$ of a prime $$p$$.

Group properties
The property listings below are for $$PSL(2,q)$$, $$q$$ a prime power.

Over a finite field
Below is a summary:

Over a finite field
Below is a summary: