Title: An Induction Theorem for the Unit Groups of Burnside Rings
of $2$-Groups

Author: Erg\" un Yal\c c\i n

Address: 
Bilkent Universitesi
Matematik Bolumu
Bilkent, Ankara, 
06800, Turkey 

Paper Status: to appear in J. Algebra.

Abstract:
Let $G$ be a $2$-group and $B(G) ^{\times}$ 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 $B(H/K)^{\times}$
to $B(G)^{\times}$ defined as the composition of inflation and
multiplicative induction maps. We prove that the product of
generalized induction maps $\prod B(H/K)^{\times} \to
B(G)^{\times}$ 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 $2^n$ with $n \geq 4$. As an
application, we give an algebraic proof for a theorem by Tornehave
\cite{Torn84} which states that tom Dieck's exponential map from
the real representation ring of $G$ to $B(G)^{\times}$ is
surjective. We also give a sufficient condition for the
surjectivity of the exponential map from the Burnside ring of $G$
to $B(G)^{\times}$.