Relative BB-Groups

  • Serge Bouc

    LAMFA-CNRS, Université de Picardie Jules Verne, 33 rue St Leu, F-80039 Amiens, France
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This paper extends the notion of BB-group to a relative context. For a finite group KK and a field F\mathbb{F} of characteristic 0, the lattice of ideals of the Green biset functor FBK\mathbb{F}B_K obtained by shifting the Burnside functor FB\mathbb{F}B by KK is described in terms of BKB_K-groups. It is shown that any finite group (L,φ)(L,\varphi) over KK admits a largest quotient BKB_K-group βK(L,φ)\beta_K(L,\varphi). The simple subquotients of FBK\mathbb{F}B_K are parametrized by BKB_K-groups, and their evaluations can be precisely determined. Finally, when pp is a prime, the restriction FBK(p)\mathbb{F}B_K^{(p)} of FBK\mathbb{F}B_K to finite pp-groups is considered, and the structure of the lattice of ideals of the Green functor FBK(p)\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 BB-Groups. Doc. Math. 24 (2019), pp. 2431–2462

DOI 10.4171/DM/730