Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

Definition

For a field k, the projective general linear group of degree two PGL(2,k) or PGL₂(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 of q elements (unique up to field isomorphism).

Arithmetic functions

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.

Function	Value	Explanation
order		| GL(2,q) | = q(q − 1)2(q + 1), and the center has order q − 1.
exponent	if p = 2, if p > 2	Elements of order p, q + 1, q − 1, all orders divide one of these.
number of conjugacy classes	q + 1 if p = 2, q + 2 if p > 2	q conjugacy classes of elements that come from elements not conjugate to their negative, 1 (if p = 2) and 2 (if p > 2) conjugacy classes coming from elements that are conjugate to their negative.