SAGS - Upper bounds for finite subgroups of algebraic groups

Consider the following setup: let G be an algebraic group over some field k. What can we say about finite subgroups of G? In particular, is it possible to bound above the orders of such subgroups? The history of such a question goes a long way back, starting from Minkowski, then Schur and others. We will briefly introduce some weak bounds given by Minkowski, then proceed to give bounds which are closely related to Schur’s idea. These bounds are not optimal for a general G, but are ’very close’ to being such, and much easier to deal with than an optimal bound. We start by introducing some notation. Let l be a prime, and vl the standard l-valuation on the l-adic numbers Ql. We will also write vl(A) to denote vl(|A|), if A is a finite set. Unless specified differently, k will be a field of characteristic different from l.