# Relative $B$-Groups

### Serge Bouc

LAMFA-CNRS, Université de Picardie Jules Verne, 33 rue St Leu, F-80039 Amiens, France

## Abstract

This paper extends the notion of $B$-group to a relative context. For a finite group $K$ and a field $\mathbb{F}$ of characteristic 0, the lattice of ideals of the Green biset functor $\mathbb{F}B_K$ obtained by shifting the Burnside functor $\mathbb{F}B$ by $K$ is described in terms of $B_K$-*groups*. It is shown that any finite group $(L,\varphi)$ over $K$ admits a *largest quotient* $B_K$-*group* $\beta_K(L,\varphi)$. The simple subquotients of $\mathbb{F}B_K$ are parametrized by $B_K$-groups, and their evaluations can be precisely determined. Finally, when $p$ is a prime, the restriction $\mathbb{F}B_K^{(p)}$ of $\mathbb{F}B_K$ to finite $p$-groups is considered, and the structure of the lattice of ideals of the Green functor $\mathbb{F}B_K^{(p)}$ is described in full detail. In particular, it is shown that this lattice is always finite.

## Cite this article

Serge Bouc, Relative $B$-Groups. Doc. Math. 24 (2019), pp. 2431–2462

DOI 10.4171/DM/730