The group of endotrivial modules in the normal case. (Q869206)

Let \(k\) be an algebraically closed field of characteristic \(p>0\), and let \(G\) be a finite group. A \(kG\)-module \(M\) is called endotrivial if \(\text{End}_k(M)\cong k\oplus Q\) where \(k\) is the trivial and \(Q\) a projective \(kG\)-module. Two endotrivial \(kG\)-modules \(M,M'\) are called equivalent if there are projective \(kG\)-modules \(R,R'\) such that \(M\oplus R\cong M'\oplus R'\). The equivalence classes of endotrivial \(kG\)-modules form a finitely generated Abelian group \(T(G)\), with respect to the tensor product. In the case where \(G\) has a normal Sylow \(p\)-subgroup \(P\), the structure of \(T(P)\) was determined by \textit{J. F. Carlson, N. Mazza} and \textit{D. K. Nakano} [J. Reine Angew. Math. 595, 93-119 (2006; Zbl 1106.20007)]. In the paper under review, the author constructs endotrivial \(kG\)-modules whose equivalence classes generate a complement \(TF(G)\) of the torsion subgroup of \(T(G)\). These modules are constructed from certain endotrivial \(kP\)-modules by first extending to their stabilizers and then tensor-inducing to \(G\). Then the author compares her generators with other generating sets for \(TF(G)\).