Some quotient Hopf algebras of the dual Steenrod algebra

Abstract. Fix a prime p, and let A be the polynomial part of the dual Steen-rod algebra. The Frobenius map on A induces the Steenrod operation fP0 on cohomology, and in this paper, we investigate this operation. We point out that if p = 2, then for any element in the cohomology of A, if one applies fP0 enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that \enough times " should be \once." The bulk of the paper is a study of some quotients of A in which the Frobenius is an isomorphism of order n. We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about fP0. The dual complete Steenrod algebra makes an appearance. 1