The Dade group of a finite group

Abstract

a b s t r a c tThe aim of this paper is to construct an equivalent of the Dade group of a p-group for an arbitrary finite group G, whose elements are equivalence classes of endo-p-permutation modules. To achieve this goal we use the theory of relative projectivity with respect to a module and that of relative endotrivial modules.

On the lifting of the Dade group

“…Unfortunately, Cap(M) may appear with a multiplicity as a direct summand of M and we shall need to avoid this. Following [Las13, Definition 5.3], we say that an endo-permutation RP -lattice M is strongly capped if it is capped and if Cap(M) has multiplicity one as a direct summand of M. We note that the class of strongly capped endo-permutation RP -lattices is closed under taking duals and tensor products (see [Las13,Lemma 5.4…”

Dade Groups for Finite Groups and Dimension Functions

“…Another approach to defining the Dade group of a finite group is given by Lassueur in [24]. There, one considers endo-p-permutation modules that are endotrivial relative to the family of non-Sylow p-subgroups of G. Such modules are called strongly capped.…”

“…Two strongly capped modules are declared equivalent if their caps are isomorphic. Lassueur defines the Dade group D(G) as the group whose elements are the equivalence classes of strongly capped endo-p-permutation kG-modules and whose group operation is induced by the tensor product (see [24,Cor.Def. 5.5]).…”

“…Lassueur's definition of the Dade group captures the differences between one dimensional representations when the Sylow p-subgroup S is normal in G. Moreover, the relationship between Lassueur's Dade group and Urfer's G-stable elements notion is completely understood: Let Υ(G) ≤ D(G) denote the subgroup of equivalence classes of strongly capped indecomposable kG-modules corresponding to one-dimensional representations of N G (S)/S under the Green correspondence. Lassueur proves that there is a short exact sequence of abelian groups 0 → Υ(G) → D(G) → D(S) G−st → 0 where the second map in the sequence is given by restriction to the Dade group of S (see [24,Theorem 7.3]). As a generalization of Lassueur's definition of strongly capped endo-p-permutation module, we define a notion of a Dade kG-module.…”

The Dade group of Mackey functors for p-groups

“…After the classification problem is solved, there appear two kinds of generalizations of endo-permutation modules. In [13] and in [20], there appear two attempts to generalize the notion of an endo-permutation module to an arbitrary finite group G. Whereas in [14], the Dade group of a fusion system over a finite p-group is constructed.…”