Conjugacy classes in $SL(2,q)$: where to find them in a particular book.

group-theorymatricesreference-request

NB: This question is tagged with . It does not require the usual type of context.

I am interested in what the conjugacy classes of $SL_2(q)$ are.

They can be found in Remark 3 of Harris et al.'s, "On Conjugacy Classes of $SL_2(q)$." Here is said remark (letting $\mathcal{F}=\Bbb F_q$):

We can describe the matrix representatives of conjugacy classes in $\mathcal{S}=SL(2,\mathcal{F})$ by four families of types ([6]):

  1. $\begin{pmatrix} r & 0 \\ 0 & r\end{pmatrix}$, where $r\in\mathcal{F}$ and $r^2=1$.
  2. $\begin{pmatrix} r & 0 \\ 0 & s\end{pmatrix}$, where $r,s\in\mathcal{F}$ and $rs=1$.
  3. $\begin{pmatrix} s & u \\ 0 & s\end{pmatrix}$, where $s\in\mathcal{F}, s^2=1$ and $u$ is either $1$ or a non-square element of $\mathcal{F}$, i.e. $u\in \mathcal{F}\setminus\{ x^2: x\in \mathcal{F}\}$.
  4. $\begin{pmatrix} 0 & 1 \\ -1 & w\end{pmatrix}$, where $w=r+r^q$ and $1=r^{1+q}$ for some $r\in \mathcal{E}\setminus\mathcal{F}$, where $\mathcal{E}$ is a quadratic extension of $\mathcal{F}$.

That is, any conjugacy class $A^\mathcal{S}$ of $\mathcal S$ must contain one of the above matrices.

Here [6] is G. James and M. Liebeck's, Representations and Characters of Groups, Cambridge mathematical textbooks, Cambridge University Press, 2001.

My university library can lend me a physical copy of that book within a week (borrowing from a different library, which I believe might cost me a couple of quid); however, I only need this bit; further: it can get me a pdf of any given chapter of the second edition, that doesn't expire.

Therefore, my question is:

In which chapter of the second edition of said book can I find a proof of the conjugacy classes of $SL(2,q)$?

This will save me time and money.

Of course, you could provide the proof here, but it wouldn't be right asking for that without more context.

The closest I could find is this MO post.

EDIT: Now that I think about it, some alternatives of where to find a proof would be welcome too!

Best Answer

Perhaps not quite in the spirit you wanted, but, still, possibly useful:

Conjugacy classes in $GL(2,\mathbb F_q)$ are the same as mappings $\mathbb F_q^2\to \mathbb F_q^2$ up to conjugation. Such linear maps have Jordan canonical forms... including the case that possibly the eigenvalues are not in the base field $\mathbb F_q$. This would be relevant for $GL(n,\mathbb F_q)$ with $n>2$.

So: diagonal matrices... Jordan-block matrices with eigenvalues in the base field... and the case that the eigenvalues are properly in a quadratic extension. The latter can be put into rational canonical form.

So, as $T$-modules, $\mathbb F_q^2$ can be $\mathbb F_q[x]/\langle x-\lambda_1\rangle\oplus \mathbb F_q[x]/\langle x-\lambda_2\rangle$, or $\mathbb F_q[x]/\langle (x-\lambda)^2\rangle$, or $\mathbb F_q[x]/\langle P\rangle$, where $P$ is a monic irreducible quadratic polynomial.

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