Semiprimitivity of Group Algebras of Infinite Simple Groups of Lie Type

Abstract

Let G be a simple group of Lie type over an infinite locally finite field F. For any field K, we prove that the group algebra $K[G]$ is semiprimitive. The argument here is a mixture of combinatorial and topological methods. Combined with earlier results, it now follows that any group algebra of an infinite locally finite simple group is semiprimitive. Furthermore, if the group is countably infinite, then the group algebra is primitive. In particular, if G is a simple group of Lie type over the field F, then $K[G]$ is a primitive ring.