A Functorial Presentation of Units of Burnside Rings

Let B-x be the biset functor over F-2 sending a finite group G to the group B-x (G) of units of its Burnside ring B(G), and let <(B-x)over cap> be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from B-x in the dual Burnside functor (F2B) over cap, or equivalently, an explicit set of generators Gs of the kernel L of the natural surjection F2B -> <(B-x)over cap>. This yields a two terms projective resolution of <(B-x)over cap>, leading to some information on the extension functors Ext(1) (-, B-x). For a finite group G, this also allows for a description of B-x (G) as a limit of groups B-x (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-x is not finitely generated, and that its dual <(B-x)over cap> is finitely generated, but not finitely presented. The last result of the paper shows in addition that Gs is a minimal set of generators of L, and it follows that the lattice of subfunctors of L is uncountable.