On the homology of elementary Abelian groups as modules over the Steenrod algebra

We examine the dual of the so-called “hit problem”, the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as a module over the Steenrod Algebra A at the prime 2. The dual problem is to determine the set of A-annihilated elements in homology. The set of A-annihilateds has been shown by David Anick to be a free associative algebra. In this note we prove that, for each k≥0, the set of k partially A-annihilateds, the set of elements that are annihilated by Sqi for each i≤2k, itself forms a free associative algebra.