Post-classification version of Jordan's theorem on finite linear groups

Abstract

Using classification of finite simple groups, I show that a finite subgroup G of GL(n)(C), where C = the complex numbers, contains a commutative normal subgroup M of index at most (n + 1)!n(alogn+b). Moreover, if G is primitive and does not contain normal subgroups that are direct products of large alternating groups, then the factor (n + 1)! can be dropped. I further show that similar statements hold also in characteristics p >/= 2, if one takes M to be an extension of a group of Lie type of characteristic p by a solvable group that has a normal p-subgroup with commutative p'-quotient. These results improve the celebrated theorems of Jordan and of Brauer and Feit.