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

Abstract

In this paper, I describe the structure of the biset functor $B^{\times }$ sending a $p$-group $P$ to the group of units of its Burnside ring $B(P)$. In particular, I show that $B^{\times }$ is a rational biset functor. It follows that if $P$ is a $p$-group, the structure of $B^{\times }(P)$ can be read from a genetic basis of $P$: the group $B^{\times }(P)$ 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.