Nilpotent elements of order \(3\) in the Green ring of the dihedral-\(2\)- groups
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.
scientific article; zbMATH DE number 687035
scientific article