An induction theorem for the unit groups of Burnside rings of 2-groups

Let G be a 2-group and B(G)× denote the group of units of the Burnside ring of G. For each subquotient H/K of G, there is a generalized induction map from B(H/K)× to B(G)× defined as the composition of inflation and multiplicative induction maps. We prove that the product of generalized induction maps ∏B(H/K)×→B(G)× is surjective when the product is taken over the set of all subquotients that are isomorphic to the trivial group or a dihedral 2-group of order 2n with n⩾4. As an application, we give an algebraic proof for a theorem by Tornehave [The unit group for the Burnside ring of a 2-group, Aarhus Universitet Preprint series 1983/84 41, May 1984] which states that tom Dieck's exponential map from the real representation ring of G to B(G)× is surjective. We also give a sufficient condition for the surjectivity of the exponential map from the Burnside ring of G to B(G)×.