# Size of maximal tori in finite simple groups of Lie type

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

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 $(3-1)^{n+1}/\gcd(n+2,3-1)=2^n$.
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}$$