Endotrivial irreducible lattices.
Let \(p\) be a prime, and let \(G\) be a finite group containing an elementary Abelian subgroup of order \(p^2\). Let \(R\) be a complete discrete valuation ring of characteristic 0 such that its fraction field \(K\) contains a \(|G|\)-th root of unity and its residue field \(F\) is an algebraically closed field of characteristic \(p\). Suppose \(M\) is an \(RG\)-lattice, i.e. \(M\) is an \(R\)-free finitely generated \(RG\)-module. Then \(M\) is called irreducible if it has no proper non-zero \(R\)-pure \(RG\)-submodule, and \(M\) is called endotrivial if \(M\otimes M^*\cong R\oplus P\) where \(R\) is the trivial \(RG\)-module and \(P\) is a projective \(RG\)-lattice. The goal of the paper under review is to analyze what the structural consequences for \(G\) are when we assume that there exists a faithful irreducible endotrivial \(RG\)-lattice. The result depends on whether or not the Fitting subgroup \(F(G)\) is central in \(G\).
scientific article; zbMATH DE number 5994291
scientific article