Maximal soluble subgroups in a parabolic subgroup of finite classical simple group - MathOverflow

Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

2012-11-24T13:11:00Z

Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ stabilizing an isotropic subspace. Is the Borel subgroup of $G$ maximal soluble in $P$ and is there any maximal soluble subgroup in $P$ besides the Borel subgroup? Here by maximal soluble I mean a maximal one among soluble subgroups of $P$.

Any feedback is appreciated!

Answer by Geoff Robinson for Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

2012-11-24T17:29:11Z

I think there are examples when the Borel subgroup is not maximal solvable in a parabolic. One can occur when $q \leq 3,$ so for example when $G = {\rm GL}(3,2)$ both the maximal parabolics are themselves solvable (isomorphic to $S_{4}$). Similarly when $p= 3$, the group ${\rm GL}(3,3)$ has a solvable maximal parabolic $P$ with unipotent radical $U$ such that $P/U \cong {\rm GL}(2,3).$ Such behavior perpetuates itself in higher ranks to give solvable parabolic subgroups strictly containing the Borel in characteristic $2$ or $3$. Later comments: Really, from an inductive point of view, it is probably best not to assume that $G$ is simple as an abstract group, and then one can argue inductively on the rank of the associated $BN$-pair.This seems to reduce us to the rank $1$ case and then it would appear that the only case to worry about is when the whole group is solvable (otherwise the Borel intersects the unique component in a maximal subgroup).

Answer by Jim Humphreys for Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

2012-11-24T21:54:00Z

EDIT: This responds to Geoff's comment (and my carelessness), but also adds a couple of other remarks.

Geoff's answer and the comments might be clarified a little as follows. You are looking at rational points of a parabolic subgroup over a finite field (where it's not important whether the algebraic group is of classical type or not). In the split (Chevalley group) case, it's easy to check orders of the various finite groups involved. Since a Levi factor of a proper parabolic group is just the product of a central algebraic torus and a lower rank semisimple group, the rational points sometimes yield a solvable group even if the original parabolic is not itself a solvable algebraic group (a Borel subgroup). This also involves the (nilpotent) group of rational points of the unipotent radical, so it gets complicated. To assemble a complete list of possibilities, look at the simple algebraic groups having solvable groups of rational points over a field of small characteristic. The torus part is already commutative, so the product and therefore the Levi subgroup of a parabolic can be solvable.

Naturally it gets more complicated to treat all types of twisted groups as well as Suzuki or Ree groups, but in principle it looks reasonable to assemble a complete list of solvable finite parabolics. (If there is enough motivation to do so.)

Concerning the determination of all maximal solvable subgroups of a finite simple group of Lie type, that's asking for a lot. The BN-structure is a valuable organizing tool, but the groups have considerably richer subgroup structure than can be identified from Lie theory alone. Even in the study of linear algebraic groups, older work of V.P. Platonov shows that it is tricky to pin down all their not necessarily connected maximal closed solvable subgroups.

[Terminology: for me language like "the Borel subgroup" is a bit jarring, since there are many conjugate choices starting with the algebraic group.]