Bouc, Serge

Relative \(B\)-Groups

Doc. Math. 24, 2431-2462 (2019)
DOI: 10.25537/dm.2019v24.2431-2462
Communicated by Henning Krause

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

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