Some quotient Hopf algebras of the dual Steenrod algebra

Fix a prime $p$, and let $A$ be the polynomial part of the dual Steenrod algebra. The Frobenius map on $A$ induces the Steenrod operation $\widetilde{\mathscr{P}}^{0}$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 $\widetilde{\mathscr{P}}^{0}$ 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 $\widetilde{\mathscr{P}}^{0}$. The dual complete Steenrod algebra makes an appearance.