Nilpotent elements of order \(3\) in the Green ring of the dihedral-\(2\)- groups
type: Item
instance of: scholarly article
title: Nilpotent elements of order \(3\) in the Green ring of the dihedral-\(2\)- groups
zbMATH Open document ID: 0830.20007
DOI: 10.1007/BF01189819
author: Hans Heldner
published in: Archiv der Mathematik
publication date: 1994-11-13T00:00:00Z
review text: Let \(G\) be a finite group and \(k\) a field of characteristic \(p\). The Green ring \(a(G):=a(kG)\) is defined to be the ring with generators the isomorphism classes \([M]\) of (finitely generated) \(kG\)-modules, and relations \([M]+[N]=[M\oplus N]\) and \([M][N]=[M\otimes N]\). If the Sylow \(p\)-subgroup of \(G\) is not cyclic (and not elementary abelian, if \(p=2)\), then \(a(G)\) has non-zero nilpotent elements. We compute an example of a nilpotent element of order 3 in the Green ring of the dihedral group \(D_8\). It is the smallest example of an infinite family of nilpotent elements, found using a program, that decomposes tensor products of dihedral modules into a direct sum of indecomposable modules [see the author's Dissertation (Zürich 1993)]. It has been shown there that the Green ring of the dihedral 2-groups contains further nilpotent elements of order 2 and 3, but it is not clear if there are nilpotent elements of order 4 or higher.
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 16N40
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 687035
zbMATH Keywords: finitely generated \(kG\)-modules
zbMATH Keywords: finite groups
zbMATH Keywords: Green rings
zbMATH Keywords: Sylow \(p\)- subgroups
zbMATH Keywords: dihedral groups
zbMATH Keywords: nilpotent elements
zbMATH Keywords: tensor products
zbMATH Keywords: direct sum of indecomposable modules
zbMATH Keywords: dihedral 2-groups
MaRDI profile type: Publication
cites work: Nilpotent elements in the Green ring
cites work: Q3999421
cites work: REPRESENTATIONS OF DIHEDRAL GROUPS OVER A FIELD OF CHARACTERISTIC 2
cites work: The indecomposable representations of the dihedral 2-groups
cites work: Nilpotent elements in representation rings
cites work: Nilpotent elements in representation rings over fields of characteristic 2
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description: scientific article; zbMATH DE number 687035