Are finite quotients of the multiplicative group of a division algebra solvable?
Recall that solvable groups are constructed by the "rule" that abelian
groups are solvable and if G is a group with a normal subgroup N such that
N and G/N are solvable, then G is solvable.
The structure of the multiplicative group of division algebras is pretty mysterious, even though they've been around for quite sometime. It is known however that they are never solvable. Recent results suggest that the following conjecture is true
CONJECTURE F.SO.Q (finite solvable quotients): Finite quotients
of the multiplicative group of a finite dimensional division
algebra are solvable.
We'll discuss Conjecture F.So.Q. and connect it to a long standing
open conjecture of Margulis and Platonov on the normal subgroup structure
of algebraic groups over number fields.