Simple Modules for Groups with Abelian Sylow 2-Subgroups are Algebraic

Let G be a finite group and let p be a prime. A module for G over a field of characteristic p is called algebraic if it satisfies a polynomial, with addition and multiplication given by direct sum and tensor product. In some sense, having this property is equivalent to the tensor structure being 'nice' for that module. In this paper we prove that if G is a group with abelian Sylow 2-subgroups, and p=2, then all simple modules for G are algebraic. We include the conjecture that this result holds for all abelian 2-blocks.