Non-isomorphic 2-groups Coincide at Most in Three Quarters of their Multiplication Tables

Let G( ∘ ) and G( * ) be two groups of the finite order n, and let d be the size of the set {(a, b) ∈ G × G;a ∘ b ≠ = a * b }. Let P and Q be Sylow 2-subgroups of G( ∘ ) and G( * ), respectively. If d is less than n2 / 4, then there exists an isomorphism ϑ: P ≃ Q, and the normalizers of P and Q have the same order.