Holomorphy of Osborn loops

TL;DR

This paper characterizes when the holomorph of an Osborn loop is itself an Osborn loop, providing necessary and sufficient conditions involving automorphisms and nuclei, and explores related algebraic structures.

Contribution

It offers a complete characterization of the holomorph of an Osborn loop being Osborn, linking automorphisms, nuclei, and isomorphisms in the loop structure.

Findings

Holomorph of an Osborn loop is Osborn iff specific automorphism conditions hold.

Conditions involving nuclei and automorphisms characterize Osborn property in holomorphs.

Structural relationships among automorphism groups and nuclei are established.

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 $(L,\cdot)$.