Dade Groups for Finite Groups and Dimension Functions

Abstract

Let G be a finite group and k an algebraically closed field of characteristic p > 0. We define the notion of a Dade kG-module as a generalization of endo-permutation modules for p-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade kG-modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group D(G) defined by Lassueur. We also consider the subgroup D Ω (G) of D(G) generated by relative syzygies ΩX , where X is a finite G-set. If C(G, p) denotes the group of superclass functions defined on the p-subgroups of G, there are natural generators ωX of C(G, p), and we prove the existence of a well-defined group homomorphism ΨG : C(G, p) → D Ω (G) that sends ωX to ΩX . The main theorem of the paper is the verification that the subgroup of C(G, p) consisting of the dimension functions of k-orientable real representations of G lies in the kernel of ΨG.