A reduction theorem for fusion systems of blocks

Abstract

Let k be an algebraically closed field of characteristic p and G a finite group. An interesting question for fusion systems is whether they can be obtained from the local structure of a block of the group algebra kG . In this paper we develop some methods to reduce this question to the case when G is a central p ′ -extension of a simple group. As an application of our result, we obtain that the ‘exotic’ examples of fusion systems discovered by Ruiz and Viruel [A. Ruiz, A. Viruel, The classification of p -local finite groups over the extra-special group of order p 3 and exponent p , Math. Z. 248 (2004) 45–65] do not occur as fusion systems of p -blocks of finite groups.