Title : The Dade group of a $p$-group
Author : Serge Bouc
Abstract :
Let $p$ be a prime number. This paper solves the question of the
structure of the group $D(P)$ of endo-permutation modules over
an arbitrary finite $p$-group $P$, that was open after Dade's
original papers in 1978, and it gives a proof of the conjectures
I proposed on this subject in 2003. This leads to a presentation
of $D(P)$ by explicit generators and relations, generalizing the
presentation obtained by Dade when $P$ is abelian.
A key result of independent interest is the explicit description
of the kernel of the natural map from the Burnside group to the
group of rational characters, in terms of the extraspecial group
of order $p^3$ and exponent $p$ if $p\neq 2$, or of all dihedral
groups of order at least 8 if $p=2$.

Status : to appear in Inventiones Mathematicae.