The Dade group of a p-group

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 ([19], [20]), and it gives a proof of the conjectures proposed in [4] and [10]. 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≠2, or of all dihedral groups of order at least 8 if p=2.