## Relative $B$-Groups

##### Doc. Math. 24, 2431-2462 (2019)
DOI: 10.25537/dm.2019v24.2431-2462

### Summary

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 $B_K$-\textit{groups}. It is shown that any finite group $(L,\varphi)$ over $K$ admits a \textit{largest quotient} $B_K$-\textit{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.

### Mathematics Subject Classification

18B99, 19A22, 20J15

### Keywords/Phrases

$B$-group, Burnside ring, biset functor, shifted functor

### References

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### Affiliation

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