Characterizing $\text{PGL}_2(\mathbb F_p)$

Question

Where can I find a description and proof of the basic structure of $\text{PGL}_2(\mathbb{F}_p)$ (number of elements with each order, conjugacy classes, etc.) which is understandable by an undergraduate who has taken a few semesters of group theory? In places such as here there is a description of the distribution of orders and a reference to the conjugacy classes, but no proof.

Answer 1

I've had two years of abstract algebra: one year in my undergraduate career and another year as a graduate; however, I've never seen an in-depth discussion of $PGL_2(\Bbb F_p)$. I've seen discussions on $PSL_n(\Bbb F_p)$ where we've proven that it's simple for most $n$ and $p$, but other than that I haven't seen matrix groups emphasized anywhere. I must be attending the wrong universities. As such, I found this to be an interesting exercise in what I've learned. Also, I don't believe it's as trivial as people seem to think it is.

First, let's recall some facts about cyclic groups. Let $Z_n$ denote the cyclic group of order $n$.

• $Z_n$ has an element of order $d$ if and only if $d\mid n$
• When $d\mid n$, the number of elements of order $d$ is $\varphi(d)$

This is the first hint that characterizing, let alone counting, the elements of a certain order of $PGL_n(\Bbb F_p)$ will be difficult. As far as I know, $\varphi(p-1)$ isn't easy to compute even when $p$ is prime.

Before we learn the elementary facts about $PGL_2(\Bbb F_p)$, let's get a handle on the elementary facts about $GL_2(\Bbb F_p)$.

• The order of $GL_2(\Bbb F_p)$ is $(p^2-1)(p^2-p)=p(p-1)^2(p+1)$. This fact is usually arrived at by counting the possible columns.
• The center of $GL_2(\Bbb F_p)$ consists of the scalar matrices. Hence the center of $GL_2(\Bbb F_p)$ has order $p-1$. This fact is usually arrived by taking an arbitrary matrix and seeing which elementary matrices it commutes with.

Thus we can see that

• $PGL_2(\Bbb F_p)$ has order $p(p-1)(p+1)$ by the 1st Isomorphism Theorem.

Now let's characterize $PGL_2(\Bbb F_2)$. Note that its order is $6$ by the above. However, note in addition that it cannot be abelian. This is because an abelian group of order 6 is cyclic. And in general if $G/Z(G)$ is cyclic, then $G$ must be abelian; however $GL_2(\Bbb F_2)$ is clearly not abelian. Thus $PGL_2(\Bbb F_2)$ is a nonabelian group of order 6 which implies that it is isomorphic to $S_3$. From now on, we assume that $p$ is an odd prime.

Now let's try to get a handle on the conjugacy classes of $GL_2(\Bbb F_p)$. This was an interesting problem. First of recall that over an algebraically closed field, to get a cross section of the conjugacy classes you can consider distinct matrices in Jordan Normal Form. We can do something similar here, but because we don't have algebraically closed we have to deal with some weird matrices. It turns out that every matrix of $GL_2(\Bbb F_p)$ is conjugate to a matrix of one of the following forms: $$\begin{bmatrix}a&0\\0&a\end{bmatrix}\quad\begin{bmatrix}a&1\\0&a\end{bmatrix}\quad\begin{bmatrix}a&0\\0&b\end{bmatrix}\quad\begin{bmatrix}d&c\\1&d\end{bmatrix}$$

where $a\neq 0$, $b\neq 0$, $a\neq b$, and $c$ is a quadratic nonresidue mod $p$. This last bit was quite a tour through my old number theory class.

Counting these kind of matrices (being careful not to double-count the matrices of the third form above), we see that $GL_2(\Bbb F_p)$ has $(p-1)(p+1)$ conjugacy classes. Going in the same order as we listed the forms above, we can list how large each of these conjugacy classes are:

• For central matrices, their conjugacy classes each have size $1$.
• For matrices of the second form, their conjugacy classes have size $(p-1)(p+1)$
• For matrices of the third form, their conjugacy classes have size $p(p+1)$
• And for the interesting matrices, their conjugacy classes have size $p(p-1)/2$

These numbers were arrived at by explicitly finding the centralizing matrices, counting them, and then finding the index in $GL_2(\Bbb F_p)$.

Finally learning something about $PGL_2(\Bbb F_p)$, we can conclude after reading this interesting question, that the number of conjugacy classes that $PGL_2(\Bbb F_P)$ has is between $p+1$ and $(p+1)(p-1)$. That's fairly weak, but I'm hoping someone with more knowledge will run with it.

Using the same forms above, we can compute the orders of some elements in $GL_2(\Bbb F_p)$.

• Central matrices of course have the same order as $a$ in $\Bbb F_p^\times$.
• Elements of the second form have order $p|a|$ where $|a|$ is the order of $a$ in $\Bbb F_p^\times$.
• Elements of the third form have order $\text{lcm}(|a|,|b|)$. In particular, this order divides $p-1$ since each $|a|$ and $|b|$ divide $p-1$.
• I haven't been able to compute the orders of the last set of matrices. I'm thinking about diagonalizing them over their splitting field $\Bbb F_{p^2}$ and seeing where that goes. However keeping Cauchy's Theorem in mind, we see that we have procured elements which have order $p$ and elements which have order dividing $p-1$, but we have yet to procure elements which assuredly divide $p+1$. Such elements will most likely come from here. Update: Indeed, the order of these elements have been discussed before both here and here. They were certainly an interesting read.

You can use this information to compute the quotient images' orders in $PGL_2(\Bbb F_p)$. Counting the elements of a certain order however seems to be undiscussed.