Modular representations of finite groups with trivial restriction to Sylow subgroups

Abstract

Let $k$ be a field of characteristic $p$. Let $G$ be a finite group of order divisible by $p$ and $P$ a $p$-Sylow subgroup of $G$. We describe the kernel of the restriction homomorphism $T(G)\to T(P)$, for $T(-)$ the group of endotrivial representations. Our description involves functions $G\to k^{\times }$ that we call weak $P$-homomorphisms. These are generalizations to possibly non-normal $P\le G$ of the classical homomorphisms $G/P\to k^{\times }$ appearing in the normal case.