Moufang semidirect products of loops with groups and inverse property extensions

TL;DR

This paper explores the structure of Moufang loops formed as semidirect products with groups, focusing on inverse property extensions influenced by cyclic groups, and provides insights and examples highlighting their complex behavior.

Contribution

It introduces a detailed analysis of Moufang semidirect products and inverse property loop extensions, addressing a specific open question and illustrating their diverse properties.

Findings

Extensions are often not well-behaved

Partial answer to Gagola III's question

Examples demonstrate complex extension behaviors

Abstract

We investigate Moufang loops which can be written as the semidirect product of a loop and a group. We also examine a particular class of loop extensions which arise as a result of a finite cyclic group acting as a group of semiautomorphisms on an inverse property loop. A question posed by Gagola III related to such extensions is partially answered, and we give several examples which illustrate that, in general, these extensions are not well-behaved.