Relative $B$-Groups

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 $(L,\phi )$ over $K$ admits a largest quotient $B_K$-group ${\beta }_K(L,\phi )$. 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.