A Functorial Presentation of Units of Burnside Rings

Abstract

Let \(B^\times \) be the biset functor over \(\mathbb {F}_2\) sending a finite group G to the group \(B^\times (G)\) of units of its Burnside ring B(G), and let \(\widehat{B^\times }\) be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from \(B^\times \) in the dual Burnside functor \(\widehat{\mathbb {F}_2B}\), or equivalently, an explicit set of generators \(\mathcal {G}_\mathcal {S}\) of the kernel L of the natural surjection \(\mathbb {F}_2B\rightarrow \widehat{B^\times }\). This yields a two terms projective resolution of \(\widehat{B^\times }\), leading to some information on the extension functors \(\mathrm{Ext}^1(-,B^\times )\). For a finite group G, this also allows for a description of \(B^\times (G)\) as a limit of groups \(B^\times (T/S)\) over sections (T, S) of G such that T/S is cyclic of odd prime order, Klein four, dihedral of order 8, or a Roquette 2-group. Another consequence is that the biset functor \(B^\times \) is not finitely generated, and that its dual \(\widehat{B^\times }\) is finitely generated, but not finitely presented. The last result of the paper shows in addition that \(\mathcal {G}_\mathcal {S}\) is a minimal set of generators of L, and it follows that the lattice of subfunctors of L is uncountable.