A nonexterior Hopf algebra and loop spaces

Let A be the Hopf algebra over Zp for a prime p given by A = Λ(xi, yi ¦ 0 ⩽ i ⩽ p − 2) ⊗ (Zp[z](zp)) (deg xi = 2(p + 1)i + 3, deg yi = 2(p + 1)(i + 1) − 1, deg z = 2(p + 1)). Kane showed that A is a minimum candidate for the mod p cohomology of a simply connected mod p finite loop space with p-torsion. In fact, if p = 2, 3, 5, then for X = G2, F4, E8, we have H∗(X; Zp) ≅ A. We prove that if p ⩾ 7, then there are no such loop spaces X.