Abstract. Gaussian polynomials are used to define bases with good multiplicative properties for the algebra K * (K) of cooperations in K-theory and for the invariants under conjugation.

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We list explicitly a minimal set of generators for the cohomology of an elementary abelian p-group, V , of rank 1 or 2, as a module over the mod p Steenrod algebra, for an odd prime p. Following Singer, we then construct a transfer map to the vector space spanned by such generators, where V now has arbitrary rank, from the homology of the Steenrod algebra. We show that this map takes images in the subspace of GL(V )-invariants and that it is an isomorphism for V having rank 1 or 2.Mathematics Subject Classification (1991): 55S10, 55Q45

We compute the subring of H*(CP∞ × CP∞; ) annihilated by the Steenrod algebra, , p being an odd prime. By calculating the subring's structure as a GL(2, )-space we may obtain information about the modular representations of that group.

Abstract. We investigate the canonical conjugation, χ, of the mod 2 dual Steenrod algebra, A * , with a view to determining the subspace, A χ * , of elements invariant under χ. We give bounds on the dimension of this subspace for each degree and show that, after inverting ξ 1 , it becomes polynomial on a natural set of generators. Finally we note that, without inverting ξ 1 , A χ * is far from being polynomial.

We say that a Hopf algebra is copolynomial if its dual is polynomial as an algebra. We re-derive Milnor's result that the mod 2 Steenrod algebra is copolynomial by means of a more general result that is also applicable to a number of other related Hopf algebras.