The paper under review constitutes an important step towards the determination of all endo-permutation modules which has recently been completed by S. Bouc [Invent. Math. 164, No. 1, 189-231 (2006; Zbl 1099.20004)]. For a field \(F\) of characteristic \(p>0\) and a finite \(p\)-group \(G\), an indecomposable finitely generated \(FG\)-module \(M\) is called an endo-permutation module if \(\text{End}_F(M)=M\otimes_FM^*\) is a permutation \(FG\)-module. It is called an endo-trivial \(FG\)-module if \(\text{End}_F(M)=F\oplus P\) where \(F\) is the trivial and \(P\) a free \(FG\)-module. The isomorphism classes of indecomposable endo-permutation \(FG\)-modules form a finitely generated Abelian group, the Dade group \(D(FG)\), and the isomorphism classes of endo-trivial \(FG\)-modules form a subgroup of \(D(FG)\) denoted by \(T(FG)\).
One of the main results of the paper under review implies that \(T(FG)\) is torsion-free if \(G\) is neither cyclic nor quaternion nor semidihedral. Another main result shows that, when \(p\) is odd, the torsion subgroup of \(D(FG)\) is isomorphic to \((\mathbb{Z}/2\mathbb{Z})^s\) where \(s\) is the number of conjugacy classes of nontrivial cyclic subgroups of \(G\).