Endotrivial irreducible lattices.
type: Item
instance of: scholarly article
title: Endotrivial irreducible lattices.
zbMATH Open document ID: 1254.20004
author: Geoffrey R. Robinson
published in: Journal of Algebra
publication date: 2012-01-04T00:00:00Z
review text: 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 \(G\).
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 5994291
zbMATH Keywords: endotrivial modules
zbMATH Keywords: endotrivial lattices
zbMATH Keywords: group rings
zbMATH Keywords: discrete valuation rings
zbMATH Keywords: finite groups
zbMATH Keywords: free finitely generated modules
zbMATH Keywords: projective lattices
zbMATH Keywords: irreducible lattices
zbMATH Keywords: Fitting subgroup
reviewed by: Frauke Bleher
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.04.007
OpenAlex ID: W2090705378
cites work: Endotrivial modules for $p$-solvable groups
cites work: On endo-trivial modules for \(p\)-solvable groups.
cites work: On simple endotrivial modules
DOI: 10.1016/J.JALGEBRA.2011.04.007
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description: scientific article; zbMATH DE number 5994291