Holomorphy of Osborn loops

Abstract

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