A positive answer to Barsotti and Carman's question on unit groups of Burnside rings of fusion systems

Abstract

Let F be a saturated fusion system on a finite 2-group S, and let B(F) be the Burnside ring of F. Barsotti and Carman proved that when S is abelian, B(F)^*=B(S)^* if and only if F=F_S(S). Barsotti and Carman asked on page 728 of their paper whether there exists a counterexample to Corollary 9.3 for a nonabelian 2-group. We have given a positive answer. Our main conclusion is that the following are equivalent for all finite 2-groups: (1)F=F_S(S); (2)B(F)^*=B(S)^*; (3)For every maximal subgroup M of S, 1-[S/M] is in B(F)^*.