Modular representations of finite groups with trivial restriction to Sylow subgroups. (Q388764)

Let \(k\) be a field of characteristic \(p>0\), let \(G\) be a finite group with Sylow \(p\)-subgroup \(P\), and let \(T(G)\) be the group of endotrivial \(kG\)-modules. The author shows that the kernel of the restriction map \(T(G)\to T(P)\) is isomorphic to the group \(A(G,P)\) of weak \(P\)-homomorphisms \(G\to k^\times\). Here a map \(u\colon G\to k^\times\) is called a weak \(P\)-homomorphism if the following three conditions are satisfied: (1) \(u(x)=1\) for all \(x\in P\); (2) \(u(x)=1\) for all \(x\in G\) such that \(P\cap x^{-1}Px\neq 1\); (3) \(u(xy)=u(x)u(y)\) for all \(x,y\in G\) such that \(P\cap y^{-1}Py\cap y^{-1}x^{-1}Pxy\neq 1\). The product in \(A(G,P)\) is defined pointwise.