Let the mod 2 Steenrod algebra, A, and the general linear group, GL_k := GL(k, F₂), act on P_k := F₂[x₁,...,x_k] with deg (x_i) = 1 in the usual manner. We prove that, for a family of some rather small subgroups G of GL_k, every element of positive degree in the invariant algebra P_k^G is hit by A in P_k. In other words, (P_k^G)⁺ ⊂ A⁺ · P_k, where (P_k^G)⁺ and A⁺ denote respectively the submodules of P_k^G and σa consisting of all elements of positive degree. This family contains most of the parabolic subgroups of GL_k. It should be noted that the smaller the group G is, the harder the problem turns out to be. Remarkably, when G is the smallest group of the family, the invariant algebra P_k^G is a polynomial algebra in k variables, whose degrees are ≤ 8 and fixed while k increases. It has been shown by Hu'ng [Trans. Amer. Math. Soc. 349 (1997), 3893-3910] that, for G = GL_k, the inclusion (P_k^GL_k)⁺ ⊂ A⁺ · P_k is equivalent to a weak algebraic version of the long-standing conjecture stating that the only spherical classes in Q₀S⁰ are the elements of Hopf invariant 1 and those of Kervaire invariant 1. © 2001 Elsevier Science.

Hung N.H.V., Nam T.N.

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