Title: Higher Conjugation Cohomology in Commutative Hopf Algebras Authors: M. D. Crossley and Sarah Whitehouse AMS Classification Numbers: 16W30, 57T05, 20C30, 20J06, 55S25 Addresses: Max-Planck-Institut fuer Mathematik, P.O. Box 7280, D-53072 Bonn, Germany. Departement de Mathematiques, Universite d'Artois - Pole de Lens, Rue Jean Souvraz, S. P. 18 - 63207 Lens, France. Email addresses: crossley\@member.ams.org whitehouse\@euler.univ-artois.fr Abstract Text: The dual Steenrod algebra can be expressed as the homotopy of a smash product of two copies of the Eilenberg-MacLane spectrum, and the conjugation arises by permutation of the two factors. This can be generalized to an action of the symmetric group $\Sigma_n$ acting on an n-1-fold tensor product of copies of the dual Steenrod algebra; this action was described by the second author in [6], purely in terms of the Hopf algebra structure. So, formally, one has a similar action for any commutative Hopf algebra. In this paper, we study the cohomology ring $H^*(\Sigma_n; A^{\otimes n-1})$, where $A$ is a graded commutative Hopf algebra. We show that for a certain class of Hopf algebras the cohomology ring is independent of the coproduct provided $n$ and $(n-2)!$ are invertible in the ground ring. Then, by choosing a sufficiently simple coproduct, we are able to deduce significant information about the $\Sigma_n$ invariants of $A^{\otimes n-1}$, including dimensions and algebra structure. In particular, we give a complete solution to the "conjugation invariants" problem for the mod $p$ dual Steenrod algebra when $p>2$.