A sectional characterization of the Dade group

Abstract

: Let k be a field of characteristic p , let P be a finite p-group, where p is an odd prime, and let D(P ) be the Dade group of endo-permutation kP -modules. It is known that D(P ) is detected via deflation-restriction by the family of all sections of P which are elementary abelian of rank ≤ 2. In this paper, we improve this result by characterizing D(P ) as the limit (with respect to deflationrestriction maps and conjugation maps) of all groups D(T /S) where T /S runs through all sections of P which are either elementary abelian of rank ≤ 3 or extraspecial of order p 3 and exponent p.

Gluing torsion endo-permutation modules

“…In fact, this is a general theorem for arbitrary biset functors which is proved in Section 7 (Theorem 7.2), while the above form for the Dade group is proved at the end of Section 8. The result also plays a crucial role in another paper [BoTh2].…”

“…Its version for the Dade group is Theorem 4.2 and will be proved at the end of the next section. It also plays a crucial role in another paper [BoTh2].…”

The Dade group of a fusion system

“…We denote by Defres Bouc and Thévenaz proved in [9,Theorem 4.7] that if p is an odd prime and X is the class of all p-groups of order at most p 3 and exponent p, then for any finite pgroup P the group homomorphism Defres X : DðPÞ ! D ðP X Þ is an isomorphism.…”

“…The purpose of the present paper is to investigate compatibility issues with respect to an arbitrary fusion system F on a finite p-group P. After a brief review on fusion systems in Section 2, we define in Section 3 the Dade group DðP; FÞ of a fusion system F on a finite p-group P and relate this to the definition of endo-ppermutation modules, due to Urfer [24]. In Sections 4 and 5, we extend some results of Bouc and Thévenaz [8], [9] to this context, and in Section 6, we describe some examples of Dade groups of fusion systems.…”

“…We define the notion of the Dade group of a fusion system and show that some of the gluing and detection results for Dade groups of finite p-groups due to Bouc and Thévenaz in [8], [9] extend to Dade groups of fusion systems. …”