The smallest subgroup whose invariants are hit by the Steenrod algebra
Let V be a k-dimensional double-struck capital F-sign2-vector space and let W be an n-dimensional vector subspace of V. Denote by GL(n, double-struck capital F-sign2) • 1k-n the subgroup of GL(V) consisting of all isomorphisms φ : V → V with φ(W) = W and φ(ν) ≡ ν (mod W) for every ν ∈ V. We show that GL(3, double-struck capital F-sign2) • 1k-3 is, in some sense, the smallest subgroup of GL(V) ≅ GL(k, double-struck capital F-sign2), whose invariants are hit by the Steenrod algebra acting on the polynomial algebra, H*(BV; double-struck capital F-sign2) ≅ double-struck capital F-sign2[x1, . . . , x k]. The result is some aspect of an algebraic version of the classical conjecture that the only spherical classes in Q0S 0 are the elements of Hopf invariant one and those of Kervaire invariant one. © 2007 Cambridge Philosophical Society.