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.

Locations

• Proceedings of the National Academy of Sciences - View
• PubMed Central - View
• Europe PMC (PubMed Central) - View - PDF
• PubMed - View