Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p$ an odd prime. Is it possible that $G$ contains a maximal torus of order $2^m$ for some positive integer $m$?

Would you please some one help me to find a reference containing size of maximal tori in finite simple groups of Lie type?

  • Clearly, yes? PSL(2,5), PSL(2,9) are examples. PSL(2,q) has a maximally split maximal torus of size (q-1)/Gcd(2,q-1), giving examples for q=5,9,17,257,65537. Basically you are looking for solutions to $\Phi(q) = 2^n$ for prime powers $q$, cyclotomic polynomials $\Phi$ and positive integers $n$. Such equations tend to lead to unsolved problems in number theory, so more refined questions might not have answers. If you don't mind working in the simply connected versions of the groups (instead of simple) then the formulas are clearer. – Jack Schmidt Apr 23 '14 at 13:50
  • @Jack Schmidt:Thanks for your tips. But a question, if we fix a simple group of Lie type namely $G$, is there any way to find for which of $\Phi$s, $\Phi(q)$ appears as a maximal torus size? – Adeleh Apr 23 '14 at 16:45
  • Yes (though I should say they are products of $\Phi$ and then divided by some the size of the center). I'll see if I can find a table. I think (but am not certain) it is more or less the factorization of the group order as given in GLS. – Jack Schmidt Apr 23 '14 at 17:09
  • Chapter 25 of Malle-Testerman's textbook has a good deal of information on this. I think if one literally has the root datum, then one can find the orders of all maximal tori, but I haven't tried doing it that way even for GL (and to me it looks very hard). It also talks about the Sylow $\Phi_d$-subgroups of the tori, which I think makes my $\Phi$ comments precise. At any rate, I'm still looking for clear tables. I think GL, SL, GU, SU, Sp should all be doable. If you work with the simple versions (or even adjoint) I think the formulas are a little worse. – Jack Schmidt Apr 23 '14 at 17:44
  • Magma has a function TwistedToriOrders to do this. – Jack Schmidt Apr 23 '14 at 17:45
up vote 0 down vote accepted

Let me know if you have a more specific question, and I can try to answer it. Here are some attempts:

Existence: For every positive integer $n\geq 2$, there is a finite simple group of Lie type with a maximal torus of order $2^n$: If $n$ is odd, then $\operatorname{PSL}(n,3)$ has a maximal torus of type $\Phi_1^{n-2} \Phi_2$ and order $(3-1)^{n-2} (3+1)=2^n$. If $n$ is even, then $\operatorname{PSL}(n+2,3)$ has a maximal torus of type $\Phi_1^{n+1}/(n,q-1)$ and order $(3-1)^{n+1}/\gcd(n+2,3-1)=2^n$.

A-type case: Maximal Tori for GL, SL, PGL, and PSL can be described fairly explicitly. Things are easiest for GL where a maximal torus is the group of units of a $n$-dimensional semisimple commutative $k$-algebra, that is, a the group of units of a direct product of field extensions of the underlying field. If the underlying field is finite, then field extensions are uniquely parameterized by their dimension. Hence we find a partition $d_1 \leq d_2 \leq \ldots \leq d_m$ of $n$, so that $n=d_1 + \ldots + d_m$. Then the maximal torus are the block diagonal matrices where the $i$th block is chosen from $\newcommand{\GL}{\operatorname{GL}}\newcommand{\GF}{\operatorname{GF}}\GL(1,q^{d_i}) \leq \GL(d_i,q)$ where nonzero elements of the field with $q^{d_i}$ elements are written as $k$-linear transformations of that field, where $k$ is the subfield with $q$ elements. Such a torus has order $(q^{d_1}-1)(q^{d_2}-1)\cdots(q^{d_n}-1)$. In SL and PGL one takes the subgroup or the quotient group, and in each case one removes a factor of $q-1$. In PSL one additionally removes a factor of $\gcd(n,q-1)$. In particular, choosing the partition $d_1=d_2=\ldots=d_{n-2}=1, d_{n-1}=2$, that is, $1+1+\ldots+1+2=n$, we get a maximal torus in GL of order $(q-1)^{n-2} (q^2-1)$ and a maximal torus in SL or PGL of order $(q-1)^{n-2}(q+1)$ and a maximal torus in PSL of order $(q-1)^{n-2}(q+1)/\gcd(n,q-1)$. Specializing this to $q=3$ and $n$ odd gives $2^{n-2} 4 / 1 = 2^n$. Similarly, choosing the partition $d_1=d_2=\ldots=d_{n+2}=1$, that is, $1+1+\ldots+1=n+2$ we get a maximal torus in GL of order $(q-1)^{n+2}$, and of SL and PGL of order $(q-1)^{n+1}$, and of PSL of order $(q-1)^{n+1}/\gcd(n,q-1)$. Choosing $q=3$ and $n$ even we get $2^{n+1}/2 = 2^n$. Since PSL(2,3) is not simple we run into trouble trying to get very tiny tori in a simple group.

Table: Here is a table for the orders of maximal tori in simply connected finite groups of untwisted Lie type of rank up to 4. $\Phi_1=q-1$, $\Phi_2=q+1$, $\Phi_3 = q^2+q+1$, $\Phi_4 =q^2+1$, $\Phi_5=q^4+q^3+q^2+q+1$, $\Phi_6=q^2-q+1$, $\Phi_8 = q^4+1$, $\Phi_{12}=q^4-q^2+1$.

$$\begin{array}{rcc|c|c|} r & G & k & T_1 & T_2 & T_3 & T_4 & T_5 \\ \hline A_1 & SL_2 & 2 & \Phi_1 & \Phi_2 \\ \hline %{}^2A_2 & SU_3 & %{}^2B_2 & Sz & %{}^2G_2 & Ree & A_2 & SL_3 & 3 & \Phi_1^2 & \Phi_1 \Phi_2 & \Phi_3 \\ \hline B_2 & Sp_4 & 5 & \Phi_1^2 & \Phi_2^2 & \Phi_1 \Phi_2 & \Phi_1 \Phi_2 & \Phi_4 \\ \hline G_2 & & 6 & \Phi_1^2 & \Phi_2^2 & \Phi_1 \Phi_2 & \Phi_1 \Phi_2 & \Phi_3 \\ &&& \Phi_6 \\ \hline %{}^2A_3 & SU_4 & %{}^2A_4 & SU_5 & %{}^3D_4 & & %{}^2F_4 & Ree & A_3 & SL_4 & 5 & \Phi_1^3 & \Phi_1 \Phi_2^2 & \Phi_1^2 \Phi_2 & \Phi_1 \Phi_3 & \Phi_2 \Phi_4 \\ \hline % {}^2 A_3 & SU_3 & S_4 B_3 & O_7 & 10 & \Phi_1^3 & \Phi_2^3 & \Phi_1\Phi_2^2 & \Phi_1^2 \Phi_2 & \Phi_1\Phi_2^2 \\ && & \Phi_1^2\Phi_2 & \Phi_1\Phi_3 & \Phi_2\Phi_4 & \Phi_1\Phi_4 & \Phi_2\Phi_6 \\ \hline C_3 & Sp_6 & 10 & \Phi_1^3 & \Phi_2^3 & \Phi_1\Phi_2^2 & \Phi_1^2 \Phi_2 & \Phi_1\Phi_2^2 \\ && & \Phi_1^2\Phi_2 & \Phi_1\Phi_3 & \Phi_2\Phi_4 & \Phi_1\Phi_4 & \Phi_2\Phi_6 \\ \hline %{}^2A_5 & SU_6 & %{}^2A_6 & SU_7 & %{}^2D_4 & O^-_8 & A_4 & SL_5 & 7 & \Phi_1^4 & \Phi_1^3\Phi_2 & \Phi_1^2\Phi_2^2 & \Phi_1^2 \Phi_3 & \Phi_1\Phi_2\Phi_4 \\ &&& \Phi_5 & \Phi_1\Phi_2\Phi_3 \\ \hline B_4 & O_9 & 20 & \Phi_1^4 & \Phi_2^4 & \Phi_1\Phi_2^3 & \Phi_1^3\Phi_2 & \Phi_1^2\Phi_2^2 \\ &&& \Phi_1\Phi_2^3 & \Phi_1^2\Phi_2^2 & \Phi_1^3\Phi_2 & \Phi_1^2\Phi_2^2 & \Phi_1^2\Phi_3 \\ C_4&Sp_8&(same)& \Phi_4^2 & \Phi_2^2\Phi_4 & \Phi_1^2\Phi_4 & \Phi_1\Phi_2\Phi_4 & \Phi_1\Phi_2 \Phi_4 \\ &&& \Phi_1\Phi_2\Phi_4 & \Phi_2^2\Phi_6 & \Phi_1\Phi_2\Phi_6 & \Phi_1\Phi_2\Phi_3 & \Phi_8 \\ \hline D_4 & O^+_8 & 13 & \Phi_1^4 & \Phi_2^4 & \Phi_1^2\Phi_2^2 & \Phi_1^2\Phi_2^2 & \Phi_1^2\Phi_2^2 \\ &&& \Phi_1\Phi_2^3 & \Phi_1^3\Phi_2 & \Phi_1^2\Phi_3 & \Phi_4^2 & \Phi_1\Phi_2\Phi_4 \\ &&& \Phi_1\Phi_2\Phi_4 & \Phi_1\Phi_2\Phi_4 & \Phi_2^2\Phi_6 & \\ \hline F_4 & & 25 & \Phi_1^4 & \Phi_2^4 & \Phi_1^3\Phi_2 & \Phi_1^3\Phi_2 & \Phi_1\Phi_2^3 \\ &&& \Phi_1\Phi_2^3 & \Phi_1^2\Phi_2^2 & \Phi_1^2\Phi_2^2 & \Phi_3^2 & \Phi_1^2\Phi_3 \\ &&& \Phi_1^2\Phi_3 & \Phi_4^2 & \Phi_1^2\Phi_4 & \Phi_2^2\Phi_4 & \Phi_1\Phi_2\Phi_4 \\ &&& \Phi_1\Phi_2\Phi_4 & \Phi_6^2 & \Phi_2^2\Phi_6 & \Phi_2^2\Phi_6 & \Phi_1\Phi_2\Phi_6 \\ &&& \Phi_1\Phi_2\Phi_3 & \Phi_1\Phi_2\Phi_3 & \Phi_1\Phi_2\Phi_6 & \Phi_8 & \Phi_{12} \\ \hline %{}^2A_7 & SU_8 %{}^2A_8 & SU_9 %{}^2D_5 & O^-_{10} %{}^2E_6 & % Rank 5+ now \end{array}$$

  • @ Jack Schmidt: I really appreciate you for your nice answer. Did you find " If n is odd, then PSL(n,3) has a maximal torus of type Φn−21Φ2 and order (3−1)n−2(3+1)=2n. If n is even, then PSL(n+2,3) has a maximal torus of type Φn+11/(n,q−1) and order (3−1)n+1/gcd(n+2,3−1)=2n" in Malle-Testerman's textbook or compute it trough a formula yourself? – Adeleh Apr 24 '14 at 3:25
  • I did those calculations by hand. I edited the answer to describe the GL case (extension fields viewed as vector spaces) and how to restrict to SL, PGL, and PSL (divide by q-1, q-1, and an additional gcd(n,q-1)). – Jack Schmidt Apr 24 '14 at 13:39

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