Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup.
type: Item
instance of: scholarly article
title: Endotrivial modules over groups with quaternion or semi-dihedral Sylow 2-subgroup.
zbMATH Open document ID: 1276.20007
DOI: 10.4171/JEMS/358
author: Nadia Mazza
author: Jacques Thévenaz
author: Jon F. Carlson
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2013-01-17T00:00:00Z
review text: In this paper, the authors determine the group \(T(G)\) of endotrivial \(kG\)-modules for an algebraically closed field \(k\) of characteristic 2 and a group \(G\) whose Sylow 2-subgroups are either semi-dihedral or generalized quaternion. Recall that a \(kG\)-module \(M\) is said to be endotrivial if its endomorphism algebra \(\mathrm{End}_k(M)\cong M^*\otimes_kM\) is, as a \(kG\)-module, stably isomorphic to the trivial simple \(kG\)-module \(k\). If \(M\) is an endotrivial \(kG\)-module, then there exists a unique (up to isomorphism) indecomposable endotrivial \(kG\)-module \(M_0\) such that \(M\) is stably isomorphic to \(M_0\). Two endotrivial \(kG\)-modules \(M\) and \(N\) are said to be equivalent if \(M_0\cong N_0\). The set \(T(G)\) is defined to be the set of all equivalence classes of endotrivial \(kG\)-modules, which becomes an Abelian group using the tensor product \(\otimes_k\). Let \(P\) be a Sylow 2-subgroup of \(G\), which is by assumption either semi-dihedral or generalized quaternion. In both cases, the authors prove that the restriction map \(T(G)\to T(P)\) is surjective. They then determine the group \(T(G)\) in detail. In the semi-dihedral case, the key ingredient in the proofs is the theory of almost split sequences. In particular, they use that there exists an almost split sequence whose middle term is stably isomorphic to the heart \(\mathrm{rad}(Q_k)/\mathrm{soc}(Q_k)\) of the projective cover \(Q_k\) of the trivial module \(k\). In the quaternion case, they first analyze the case when the unique involution \(z\) of \(P\) is central in \(G\). For the general case, they use that the centralizer of \(z\) is a strongly 2-embedded subgroup of \(G\). They moreover show that there are always torsion endotrivial modules which are uniserial.
reviewed by: Frauke Bleher
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 6126559
zbMATH Keywords: groups of endotrivial modules
zbMATH Keywords: exotic endotrivial modules
zbMATH Keywords: semi-dihedral groups
zbMATH Keywords: generalized quaternion groups
zbMATH Keywords: almost split sequences
zbMATH Keywords: syzygies
zbMATH Keywords: Sylow 2-subgroups
zbMATH Keywords: stable isomorphisms
describes a project that uses: Magma
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/jems/358
OpenAlex ID: W2046993141
description: scientific article; zbMATH DE number 6126559