Dade Groups for Finite Groups and Dimension Functions

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 $Ω_X$, 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 $ω_X$ of $C(G,p)$, and we prove the existence of a well-defined group homomorphism $Ψ_G:C(G,p)\to D^Ω(G)$ that sends $ω_X$ to $Ω_X$. 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 $Ψ_G$.

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