I want to prove the following statement:
All maximal tori in a solvable algebraic group are conjugate to each other.
I use the following facts:
- Theorem. Let $G$ be an irreducible solvable algebraic group and $T$ a torus complementary to its unipotent radical $U$. Then any semisimple element of $G$ is conjugate to some element of $T$.
- Lemma. Any torus has elements which are not contained in any of its proper algebraic subgroups. I call ad-hoc these element "bagels".
My idea is as follows:
Let $t_1 \in T_1$ be a bagel of a maximal torus $T_1$. It is semisimple. Then by Theorem 1 it is conjugate to some element of another torus, say $T_2$. Then $g t_1 g^{-1} \in T_2$. Since $g t_1 g^{-1}$ is also a bagel (I say that it is true because conjugation is an automorphism), then $T_2$ is maximal (since otherwise, if $T_2$ is a subgroup of another torus $T_3$ (i.e. $T_2 \subset T_3$) then $g t_1 g^{-1}$ is not a bagel since it is contained in a proper subgroup of a torus). We then can apply the same argument on bagels of $T_2$ and every other maximal torus.
I would like to know rather my argument is valid and if there are any gaps I need to close.
Added in Edit:
One gap I have found: I need to reduce the case of solvable algebraic group $G$ with maximal torus $T_1$ to a case of irreducible algebraic $H$ with $T$ is complementary to $U(H)$ = unipotent radical of $H$, in order to apply Theorem 1. How do I fix it?
I read in some book that if $U$ is the unipotent radical of $G$ then $T$, its complementary torus, is a maximal torus and the group $G$ containing it is irreducible. We also have $G = U \rtimes T$, so I think this close the first gap I mentioned.
Added in Edit 2:
I think I managed to proved this elementwise. However, I need to prove a stronger result:
let $T$ be the maximal torus complementary to the unipotent radical $U$. Then for every other maximal torus, $T'$, there exists $g \in G$ such that $ g T' g^{-1} = T $ (that is, $g$ is global to all the elements of the torus).
Added in Edit 3:
The globality of the conjugating element follows from that a torus is a group of commuting semisimple matrices and therefore can be simultaniously diagonalized.