# The Roquette category of finite $p$-groups

### Serge Bouc

Université de Picardie - Jules Verne, Amiens, France

## Abstract

Let $p$ be a prime number. This paper introduces the *Roquette category* $\mathcal{R}_p$ of finite $p$-groups, which is an additive tensor category containing all finite $p$-groups among its objects. In $\mathcal{R}_p$, every finite $p$-group $P$ admits a canonical direct summand $\partial P$, called *the edge* of $P$. Moreover $P$ splits uniquely as a direct sum of edges of *Roquette $p$-groups*, and the tensor structure of $\mathcal{R}_p$ can be described in terms of such edges.

The main motivation for considering this category is that the additive functors from $\mathcal{R}_p$ to abelian groups are exactly the *rational $p$-biset functors*. This yields in particular very efficient ways of computing such functors on arbitrary $p$-groups: this applies to the representation functors $R_K$, where $K$ is any field of characteristic 0, but also to the functor of units of Burnside rings, or to the torsion part of the Dade group.

## Cite this article

Serge Bouc, The Roquette category of finite $p$-groups. J. Eur. Math. Soc. 17 (2015), no. 11, pp. 2843–2886

DOI 10.4171/JEMS/573