# About the maximal abelian subgroups of $SL_2(F)$

Recently, I began to read a lecture notes, download from internet. The lecture is about finite simple groups. The first interesting thing in this note is about the maximal abelian subgroups $A$ of $G=SL_2(F)$, where $F$ is a field of characteristic $p\ge 0$: 1) $A$ is a unipotent radical of the Borel subgroup; 2) $A$ is a maximal split torus; 3) $A$ is a maximal non-split torus.

The first two types are easy to understand by matrix, but the third type is complex to me, and the lecture is not clear about this. Is there anyone give me some detail for this, or give me some references for this?

I also find that there are maybe some trick in the computation of matrix, which can not be found in the course of linear algebra. For example, is there special way to computing $[A,B]$ or oder of $A$ for $A, B \in GL_n(F)$?

Thank you very much!

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You can read $SL_2(\mathbb{F}_q)$ from C.Bonnafé, there's a description of the non-split torus in this case. –  th.ng Mar 26 '12 at 5:16
I think you meant 1) the unipotent radical of a Borel. –  Marc Palm Mar 26 '12 at 8:43
I think, it is best to start with the rational canonical form to understand $GL(n)$... but that's probably a matter of taste. –  Marc Palm Mar 26 '12 at 8:44
Yes, type 1) is the unipotent radical of a Borel. Please tell me some references which talk about the rational canoical form. I have not know much about this, and I can not find this in textbooks I have. –  Wei Zhou Mar 26 '12 at 9:27
Note that the rational form holds in arbitrary charactersitic and imperfect fields as well. –  Marc Palm Mar 27 '12 at 6:15

Here, is another point of view.

Take any elliptic element $\gamma$, i.e. with irreducible characteristic polynomial, then the centralizer is a maximal non split torus.

In fact, the centralizer is $F[\gamma]^\times$.

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Yes, in the lecture, the non-split torus is define in this way. But I am not familiar with this, and I can not find reference talking about this. It will be helpful if you tell me references dealing in this way. –  Wei Zhou Mar 26 '12 at 9:23
Google is your friend, there is much material about the rational canonical form freely available on the internet, even videos. Also look here: mathoverflow.net/questions/78312/… –  Marc Palm Mar 26 '12 at 9:47
–  Marc Palm Mar 26 '12 at 9:53
Thanks. I think I need more time to understanding them. –  Wei Zhou Mar 26 '12 at 14:47

The construction of a maximal non-split torus has always felt trickier to me than the other two you mention, but it isn't too hard to understand. I learned most of this by reading Cédric Bonnafé's text Representations of $SL_2(\mathbb{F}_q).$

If we take $K / F$ to be a field extension of degree two, then we can regard $K$ as a two dimensional vector space over $F$, and as multiplication by elements of $K$ is an $F$-linear operation, by choosing a basis of $K$ as a vector space over $F$ we get that $K^\times$ is isomorphic to some subgroup of $GL_2(F)$. Through this isomorphism, the trace and norm of the field extension correspond respectively to the trace and determinant of the matrices. This shows that the elements of $K$ which are of norm 1 over $F$ can be identified with a subgroup of $SL_2(F)$, this subgroup is a maximal non-split torus.

For me the problem with this construction is that it doesn't necessarily make it easy to see what the matrices in this non-split torus look like. What helps me is to try and understand them through analogy with $SL_2(\mathbb{R})$, because there I know a field extension of degree 2 that I understand well. Using 1 and $i$ as our basis for $\mathbb{C}$ and tracing through the isomorphisms there gives us that our non-split torus is $SO_2(\mathbb{R})$ as a subgroup of $SL_2(\mathbb{R})$ in the usual way. While I feel I can get a good mental grasp on this group, it still feels quite a bit more complicated to me than the standard split torus.

I hope you find this useful, It is treated fairly well in Bonnafé's aforementioned text, all in chapter 1, section 1.1.2. There are some errors in some later calculations in the text, but the section on the non-split torus is good, and there is a good exercise on calculating a good basis for $\mathbb{F}_{q^2}$ over $\mathbb{F}_q$ for calculating the matrices you get in the non-split torus.

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Thank you very much! You answer is helpful. But more problems come to me. I see that such subgroup is abelian. But I don't see why it is maximal abelian. It seems that it is good form for computation by the following comments of Paul Broussous. –  Wei Zhou Mar 26 '12 at 6:57
It's easier to see the maximality of $K^\times$ as an abelian subgroup of $GL_F(K)$. In fact if we select any element $\xi \on K^\times \setminus F^\times$ then as $K = F \oplus F\xi$ as an $F$-vector space. It is easy to show that any operator in $GL_F(K)$ that commutes with $\xi$, will act as a scalar on all elements of the form $a + b \xi$ with $a, b \in F$, and therefore on all of $K$. Bonnafé does the calculation in section 1.3 of his text. –  Ted Nitz Mar 26 '12 at 13:48
I clearly should have coffee before commenting, that should be $\xi \in K^\times \setminus F^\times$. –  Ted Nitz Mar 26 '12 at 13:49

If the characteristic of the field is not $2$, the description of non-split maximal tori is straightforward. Take an element $u$ of the base field $F$ which is not a square. Then you get such a torus as the set of matrices

$$\left( \begin{array}{cc} x & uy \newline y & x \end{array} \right)$$

where $x^2 -uy^2 =1$. Any maximal non-split torus is conjugate to this torus for some well chosen $u$.

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Thanks. Please give some reference in which I can find this. I want to find other relate information. For example, how about the normalizer of the non-split maximal tori. All the new books of group theory seems no interesting about this. –  Wei Zhou Mar 26 '12 at 7:02