Element structure of projective special linear group of degree two over a finite field

This article describes the element structure of projective special linear group of degree two over a finite field of order $$q$$ and characteristic $$p$$. Some aspects of this discussion, with suitable infinitary analogues of cardinality, carry over to infinite fields and fields of infinite characteristic.

Conjugacy class structure
See also element structure of special linear group of degree two.

Number of conjugacy classes
As we know in general, number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size, the degree of this PORC function is one less than the degree of matrices, and we make cases based on the congruence classes modulo the degree of matrices. Thus, we expect that the number of conjugacy classes is a PORC function of the field size of degree 2 - 1 = 1, and we need to make cases based on the congruence class of the field size modulo 2. Moreover, the general theory also tells us that the polynomial function of $$q$$ depends only on the value of $$\operatorname{gcd}(n,q-1)$$, which in turn can be determined by the congruence class of $$q$$ mod $$n$$ (with $$n = 2$$ here).

Case where $$p = 2$$
In this case, the natural surjective map from the special linear group of degree two to the projective special linear group of degree two is an isomorphism, so the conjugacy class structure of both groups is the same. Details below: