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 $\alpha$ and $\phi$ 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 $x\alpha (yz\cdot x\phi^{-1})= x\alpha (yx^\lambda\cdot x) \cdot zx\phi^{-1}$. 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, $(x\alpha\cdot x^{\rho})x=x\alpha$, $x(x^{\lambda}\cdot x\phi^{-1})=x\phi^{-1}$ and every pair of automorphisms in $A(L)$ is nuclear (i.e. $x\alpha\cdot x^{\rho},x^{\lambda}\cdot x\phi\in N(L,\cdot )$). It is shown that if $H(L)$ is an Osborn loop, then $A(L,\cdot)= \mathcal{P}(L,\cdot)\cap\Lambda(L,\cdot)\cap\Phi(L,\cdot)\cap\Psi(L,\cdot)$ and for any $\alpha\in A(L)$, $\alpha= L_{e\pi}=R^{-1}_{e\varrho}$ for some $\pi\in \Phi(L,\cdot)$ and some $\varrho\in \Psi(L,\cdot)$. Some commutative diagrams are deduced by considering isomorphisms among the various groups of regular bijections (whose intersection is $A(L)$) and the nucleus of $(L,\cdot)$.