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

This article gives the element structure of the projective general linear group of degree two over a finite field. Some of the structure information generalizes to infinite fields.

See also element structure of general linear group of degree two, element structure of special linear group of degree two, and element structure of projective special linear group of degree two.

We denote by $$q$$ the order (size) of the field and by $$p$$ the prime number that is the characteristic. Note that $$q$$ must be a power of $$p$$.

The order of the group is $$q^3 - q$$.

Conjugacy class structure
As we know in general, number of conjugacy classes in projective general 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).