The functor of units of Burnside rings for $p$-groups

Abstract

In this paper, I describe the structure of the biset functor B^× sending a p -group P to the group of units of its Burnside ring B(P) . In particular, I show that B^× is a rational biset functor. It follows that if P is a p -group, the structure of B^×(P) can be read from a genetic basis of P : the group B^×(P) is an elementary abelian 2-group of rank equal to the number isomorphism classes of rational irreducible representations of P whose type is trivial, cyclic of order 2, or dihedral.