# Element in finite number of Borel subgroups

Let G is a linear algebraic group over algebraic closed field, B is an Borel subgroup of G. Does there exist g$$\in$$G which is only in a finite numbers of conjugates of B (they are also Borel subgroups) ?

I choose this version of condition from the book:Tauvel, Patrice, and W. T. Rupert. Lie algebras and algebraic groups 28.2.1.

It appears in the lemma before the density theorem of Borel subgroups, but I do not see any book do this for Borel subgroups directly, they all do it for Cartan subgroups, choosing a semisimple regular element of the unique maximal torus, but Borel subgroups are bigger, maybe they have more intersections.

The similar question is at here with no answer: https://math.stackexchange.com/questions/3113958/element-in-finite-number-of-borel-subgroups

• Regular semisimple elements have this property; the only Borel subgroups containing them will be Weyl group conjugates of a fixed one. Mar 8 '19 at 7:33
• Thanks. Weyl groups appear later in many books, maybe the question should be changed, can we get the density theorem from this direct way, without vicious circle? Mar 8 '19 at 14:18
• More generally, regularity (without assuming semisimplicity) can be defined as belonging to only a finite number of Borel subgroups (or, equivalently, having minimal-dimensional centraliser). Jan 30 '20 at 20:11

• There's no reason to assume $G$ semisimple/reductive. Indeed, Borel subgroups contain the solvable radical $R$, and more precisely are the inverse images of Borel subgroups of the corresponding semisimple quotient $G/R$. Hence the question immediately reduces to the semisimple case.