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Abstract. The mod 2 Steenrod algebra A and Dyer-Lashof algebra R have both striking similarities and differences arising from their common origins in "lower-indexed" algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra K, whose module actions are equivalent to, but quite different from, those of A and R. The exact relationships emerge as "sheared algebra bijections", which also illuminate the role of the cohomology of K. As a bialgebra, K * has a particularly attractive and potentially useful structure, providing a bridge between those of A * and R * , and suggesting possible applications to the Miller spectral sequence and the A structure of Dickson algebras.

(Volume 129 (2000), 263–275)It is with very great regret that we record the death of one of the authors, Frank Peterson, shortly before the appearance of the paper. The Authors and Editor wish to express their great personal and professional respect for him.

The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A, and is isomorphic to the mod two cohomology of BO , the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., minimal generators and minimal relations.From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2 m − 1) that classify finite dimensional vector bundles, and the connected covers of BO . The presentations then show that certain of these unstable A-algebras coalesce to produce the Dickson algebras of general linear group invariants, and we speculate about possible related topological realizability.Our methods also produce a related simple minimal A-module presentation of the cohomology of infinite dimensional real projective space, with filtered quotients the unstable modules F (2 p − 1) /AA p−2 , as described in an independent appendix.

The Dickson algebra Wn+1 of invariants in a polynomial algebra over F2 is an unstable algebra over the mod 2 Steenrod algebra A, or equivalently, over the Kudo-Araki-May algebra K of "lower" operations. We prove that Wn+1 is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyze the interplay of actions over A and K to characterize unstable cyclic modules with trivial action by the subalgebra An−2 on a fundamental class in degree 2 n − a, thereby verifying unstable instances of a conjectured basis for A/AAn−2. This involves a new family of left ideals Ia in K, which play the role filled by the ideals AAn−2 in the Steenrod algebra.

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A new action of the Kudo-Araki-May algebra on the dual of the symmetric algebras, with applications to the hit problem DAVID PENGELLEY FRANK WILLIAMSThe hit problem for a cohomology module over the Steenrod algebra A asks for a minimal set of A-generators for the module. In this paper we consider the symmetric algebras over the field F p , for p an arbitrary prime, and treat the equivalent problem of determining the set of A -primitive elements in their duals. We produce a method for generating new primitives from known ones via a new action of the Kudo-ArakiMay algebra K, and consider the K-module structure of the primitives, which form a sub K-algebra of the dual of the infinite symmetric algebra. Our examples show that the K-action on the primitives is not free. Our new action encompasses, on the finite symmetric algebras, the operators introduced by Kameko for studying the hit problem.