On the Hopf Algebra Structure of the mod 2 Cohomology of a Finite $H$-Space

The action of the Steenrod algebra on the cohomology of a finite //-space has placed several restrictions on the possible spaces which can be finite //"-spaces. It is now known, for example, that the loop space of a finite //-space has no homology torsion. We continue this study here by showing that in the mod 2 cohomology of a finite //-space, every generator of degree 4/e + l is in the image of Sq. This result has several consequences. For example, if an integer n has dyadic expansion n=2+ +2