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

Abstract

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)×. © 2005 Elsevier Inc. All rights reserved

Tornehave Morphisms I: Resurrecting the Virtual Permutation Sets Annihilated by Linearization

“…They are discussed in Bouc [8,Sections 5,7], Yalçın [18,Section 3], Yoshida [19,Sections 2,3]; see also a review in [3,Section 10]. For the purposes of the present article, though, we need invoke only Yoshida's result [19, 3.5], which asserts that the maps die G commute with the five elemental maps.…”

“…The first triangle, expressing the equality exp = die lin, is discussed in Yoshida [19] and Yalçın [18]. Let us review a few features that we shall be needing later.…”

“…We draw from a stream of literature that includes Bouc-Yalçın [10], Bouc [8,9], Yalçın [18]. We also have a view towards two further articles on Tornehave morphisms, [3,4].…”

Fibred permutation sets and the idempotents and units of monomial Burnside rings

“…Yoshida [27] later vindicated the prediction as to the "2-primary structure." Tornehave [24] and Yalçın [25] have shown that the 2-group case has rich special features. But, by the Odd Order Theorem, B(G) * captures nothing at all when |G| is odd.…”

The functor of units of Burnside rings for $p$-groups

“…Recently, E. Yalçin wrote a very nice paper ( [14]), in which he proves an induction theorem for B × for 2-groups, which says that if P is a 2-group, then any element of B × (P ) is a sum of elements obtained by inflation and tensor induction from sections (T, S) of P , such that T /S is trivial or dihedral.…”