The smallest subgroup whose invariants are hit by the Steenrod algebra

Let V be a k-dimensional ${\mathbb{F}_2}$ -vector space and let W be an n-dimensional vector subspace of V. Denote by GL(n, ${\mathbb{F}_2}$ ) • 1 k-n the subgroup of GL(V) consisting of all isomorphisms ϕ:V → V with ϕ(W) = W and ϕ(v) ≡ v (mod W) for every v ∈ V. We show that GL(3, ${\mathbb{F}_2}$ ) • 1 k-3 is, in some sense, the smallest subgroup of GL(V)≅ GL(k, ${\mathbb{F}_2})$ , whose invariants are hit by the Steenrod algebra acting on the polynomial algebra, ${\mathbb{F}_2})\cong{\mathbb{F}_2}[x_{1},\ldots,x_{k}]$ . The result is some aspect of an algebraic version of the classical conjecture that the only spherical classes in Q 0 S 0 are the elements of Hopf invariant one and those of Kervaire invariant one.