The functor of units of Burnside rings for p-groups

Serge Bouc (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

In this note 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.

Mots clés: Burnside ring; unit; biset functor

Domaines: Théorie des groupes [math.GR]

Serge Bouc. The functor of units of Burnside rings for p-groups. 2006. ⟨hal-00087819⟩