| Abstract: | Suppose that (G) is a finite group such that (mathrm{SL }(n,q)subseteq G subseteq mathrm{GL }(n,q)), and that (Z) is a central subgroup of (G). Let (T(G/Z)) be the abelian group of equivalence classes of endotrivial (k(G/Z))-modules, where (k) is an algebraically closed field of characteristic (p) not dividing (q). We show that the torsion free rank of (T(G/Z)) is at most one, and we determine (T(G/Z)) in the case that the Sylow (p)-subgroup of (G) is abelian and nontrivial. The proofs for the torsion subgroup of (T(G/Z)) use the theory of Young modules for (mathrm{GL }(n,q)) and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules. |
