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

### Serge Bouc

Université de Picardie - Jules Verne, Amiens, France

## 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.

## Cite this article

Serge Bouc, The functor of units of Burnside rings for $p$-groups. Comment. Math. Helv. 82 (2007), no. 3, pp. 583–615

DOI 10.4171/CMH/103