Cyclic extensions of Moufang loops induced by semi-automorphisms

TL;DR

This paper generalizes the construction of group extensions to Moufang loops using semi-automorphisms, showing how cyclic extensions depend on the normal subloop and semi-automorphism actions.

Contribution

It introduces a method to construct cyclic extensions of Moufang loops via semi-automorphisms, expanding the theory beyond traditional automorphism-based extensions.

Findings

Extension depends on semi-automorphism actions

Cyclic group order coprime to three is crucial

Binary operation determined by semi-automorphism structure

Abstract

It is well known that if a group G factorizes as G = NH where H\leq G and N is normal in G then the group structure of G is determined by the subgroups H and N, the intersection of N with H and how H acts on N with a homomorphism f : H -> Aut(N). Here we generalize the idea by creating extensions using the semi-automorphism group of N. We show that if G=NH is a Moufang loop, N is a normal subloop, and H=<u> is a finite cyclic group of order coprime to three then the binary operation of G depends only on the binary operation of N, the intersection of N with H, and how u permutes the elements of N as a semi-automorphism of N.