Holomorphy of Osborn loops

Let (L, ·) be any loop and let A(L) be a group of automorphisms of (L, ·) such that α and φ are elements of A(L). It is shown that, for all x, y, z ∈ L, the A(L)-holomorph (H, ○) = H(L) of (L, ·) is an Osborn loop if and only if xα(yz · xφ⁻¹) = xα(yx^λ · x) · zxφ⁻¹. Furthermore, it is shown that for all x ∈ L, H(L) is an Osborn loop if and only if (L, ·) is an Osborn loop, (xα· x^ρ)x = xα, x(x^λ · xφ⁻¹) = xφ⁻¹ and every pair of automorphisms in A(L) is nuclear (i.e. xα·x^ρ, x^λ ·xφ ∈ N(L, ·)). It is shown that if H(L) is an Osborn loop, then A(L, ·) = 𝒫(L, ·)∩Λ(L, ·)∩Φ(L, ·)∩ Ψ(L, ·) and for any α ∈ A(L), α=Leπ=Reϱ−1$\alpha = L_{e\pi } = R_{e\varrho }^{ - 1}$ for some π ∈ Φ(L, ·) and some ϱ ∈ Ψ(L, ·). Some commutative diagrams are deduced by considering isomorphisms among the various groups of regular bijections (whose intersection is A(L)) and the nucleus of (L, ·).