The book under review gives a survey (usually without proofs) of the extensive literature developed in the last 40 years on endotrivial and related modules for finite groups. Endotrivial modules are of interest since they induce self-equivalences of the stable module category of a finite group, and since they appear as sources of simple modules over \(p\)-soluble groups.
Let \(k\) be a field of characteristic \(p>0\), let \(G\) be a finite group with Sylow \(p\)-subgroup \(S \ne 1\), and let \(N := N_G(S)\). A finitely generated \(kG\)-module \(M\) is called endotrivial if \(\mathrm{End}_k(M)\) is isomorphic to the trivial module in \(\mathrm{stmod}(kG)\), the stable module category of the group algebra \(kG\). The tensor product of two endotrivial \(kG\)-modules is again endotrivial. This defines a group structure on \(T(G)\), the set of \(\mathrm{stmod}(kG)\)-isomorphism classes of endotrivial \(kG\)-modules. By a result of Puig, \(T(G)\) is a finitely generated abelian group. Thus \(T(G) = TT(G) \oplus TF(G)\) where \(TT(G)\) is the torsion subgroup of \(T(G)\), and \(TF(G)\) is a free abelian subgroup of \(T(G)\).
A large part of the book under review is concerned with the structure of \(T(G)\). In general, the rank of \(TF(G)\) is equal to the number of connected components of \(\mathcal{E}_{\ge 2}(G)/G\), the poset of conjugacy classes of noncyclic elementary abelian \(p\)-subgroups of \(G\). The structure of \(TT(G)\) is more involved. First, it is known that \(TT(S) = 0\) unless \(S\) is cyclic with \(|S| > 2\), quaternion or semidihedral. If \(S\) is semidihedral, or cyclic with \(|S| > 2\), then \(TT(S) \cong \mathbb{Z}/2\). If \(S\) is quaternion and \(|S| > 8\) then \(TT(S) \cong \mathbb{Z}/4 \oplus \mathbb{Z}/2\). If \(S\) is quaternion and \(|S| = 8\) then \(TT(G) \cong \mathbb{Z}/4 \oplus X\) where \(X \cong \mathbb{Z}/2\) if \(k\) contains a primitive cube root of 1, and \(X=0\) otherwise.
Next, there is always a short exact sequence \(0 \longrightarrow N/N'S \longrightarrow T(N) \longrightarrow T(S)^N \longrightarrow 0\); in particular, \(TT(N) \cong N/N'S\) whenever \(TT(S) = 0\). If \(S\) is cyclic with \(|S| > 2\), quaternion or semidihedral then \(T(S)^N = T(S)\).
The kernel \(K(G)\) of the restriction map \(T(G) \longrightarrow T(S)\) is an important subgroup of \(TT(G)\). It is known that \(G/G'S\) embeds into \(K(G)\), and that \(K(G)\) embeds into \(N/N'S\). A recent result by Grodal shows that \(K(G) \cong N/\rho^\infty(S)\) where \(\rho^\infty(S)\) is a specific normal subgroup of \(N\). This confirms a conjecture by Carlson and Thévenaz. The author presents several cases in which \(K(G)\) has been computed: groups with cyclic, dihedral, quaternion or semidihedral Sylow subgroups, \(p\)-soluble groups, symmetric and related groups, sporadic groups and their covering groups, and finite groups of Lie type.
Other topics treated in the book include the construction of explicit generators of \(T(G)\), the lifting of endotrivial \(kG\)-modules to endotrivial \(\mathcal{O}G\)-modules where \(\mathcal{O}\) is a complete discrete valuation ring with residue field \(k\), the position of endotrivial modules in the Auslander-Reiten quiver of \(kG\), results by Robinson and Navarro on simple endotrivial modules, Balmer’s description of \(K(G)\) by weak homomorphisms, and Grodal’s approach to \(K(G)\) via homotopy theory.