Finite projective planes and sharply 2-transitive subsets of finite groups

A sharply 2-transitive subset of a permutation group G acting on a set Σ is a subset R of G with the properties (1) if α, β, γ, δ ∈ Σ, α ≠ β, γ ≠ δ, R contains a unique member r with r(α) = γ, r(β) = δ, (2) 1 ∈ R, (3) the relation ~ defined on R by r ~ s if r = s or r −1 s fixes no symbol of Σ is an equivalence relation and each equivalence class is sharply transitive on Σ, that is, if α, γ ∈ Σ each class contains exactly one member r with r(α) = γ.