On the Tensor Products of Modules for Dihedral 2-Groups

The only groups for which all indecomposable modules are ‘knowable’ are those with cyclic, dihedral, semidihedral, and quaternion Sylow p-subgroups. The structure of the Green ring for groups with cyclic and V4 Sylow p-subgroups are known, but no others have been determined. Of the remaining groups, the dihedral 2-groups have the simplest module category but yet the tensor products of any two indecomposable modules are not known. In this article, we will prove that most non-periodic modules have a complicated tensor structure. Following Alperin in [1], we define a module to be algebraic if it satisfies a polynomial with integer coefficients, where addition and multiplication are given by the direct sum and the tensor product. It is clear that a module M is algebraic if and only if there are only finitely many isomorphism types of indecomposable summand in the collection of modules M⊗n for all n > 0. Examples include all projective modules, more generally all trivial source modules, and all simple modules for p-soluble groups [7] and groups with abelian Sylow 2-subgroups [6].