Reduction theorems for groups of matrices Original Research Article

We show that if k is an algebraically closed field and G a not necessarily connected reductive linear algebraic group over k, then G(k) is solvable, nilpotent or abelian if and only if every finite subgroup of G(k) is solvable, nilpotent or abelian respectively. We also obtain the analogous result for compact subgroups of image.