On the Group of $p$-Endotrivial $kG$-Modules

Abstract

In this paper, we define a group $T_p(G)$ of $p$-endotrivial $kG$-modules and a generalized Dade group $D_p(G)$ for a finite group $G$. We prove that $T_p(G)\cong T_p(H)$ whenever the subgroup $H$ contains a normalizer of a Sylow $p$-subgroup of $G$, in this case, $K(G)\cong K(H)$. We also prove that the group $D_p(G)$ can be embedded into $T_p(G)$ as a subgroup.