The Dade group of a metacyclic \(p\)-group
type: Item
instance of: scholarly article
title: The Dade group of a metacyclic \(p\)-group
zbMATH Open document ID: 1029.20006
DOI: 10.1016/S0021-8693(03)00328-4
author: Nadia Mazza
published in: Journal of Algebra
publication date: 2003-08-20T00:00:00Z
review text: Let \(P\) be a finite \(p\)-group and \(k\) an algebraically closed field of characteristic \(p\). A \(kP\)-module \(M\) is endo-permutation, if \(\text{End}_k(M)\) is a permutation \(kP\)-module. In this case \(M\) has a unique (up to isomorphism) indecomposable summand \(M_0\) with vertex \(P\), called the cap of \(M\). Two endo-permutation modules are called equivalent if they have isomorphic caps. The equivalence classes of endo-permutation \(kP\)-modules form an Abelian group \(D(P)\) with respect to the tensor product over \(k\) and taking the \(k\)-dual. The main result of this paper is the computation of \(D(P)\) when \(P\) is metacyclic. It is also shown that in this case, the torsion subgroup of \(D(P)\) has order \(2\) (or also \(4\) in the case \(p=2\)).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 19A22
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 1967901
zbMATH Keywords: metacyclic \(p\)-groups
zbMATH Keywords: endo-permutation modules
zbMATH Keywords: Dade groups
zbMATH Keywords: indecomposable summands
zbMATH Keywords: caps
zbMATH Keywords: tensor products
zbMATH Keywords: vertices
reviewed by: Andrei Marcus
MaRDI profile type: Publication
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full work available at URL: s0021-8693(03)00328-4
OpenAlex ID: W2073485676
description: scientific article; zbMATH DE number 1967901