Power graphs of Moufang loops

TL;DR

This paper extends the understanding of power graphs from groups to Moufang loops, showing that isomorphic undirected power graphs imply isomorphic directed power graphs for these loops, and introduces generalized octonion loops.

Contribution

It proves that Moufang loops with isomorphic undirected power graphs have isomorphic directed power graphs, extending known group results to loops, and introduces generalized octonion loops.

Findings

Isomorphic undirected power graphs imply isomorphic directed power graphs for Moufang loops.

Introduction of generalized octonion loops with quaternion-like behavior.

Extension of group power graph results to the context of Moufang loops.

Abstract

Power graphs of both groups and semigroups have been widely studied. While the power graph of a quasigroup can be defined analogously to that of a group, power graphs of quasigroups and loops have thus far been little studied. In this paper we begin transferring results on the power graphs of groups to the context of loops by addressing a question posed by Peter Cameron: if two Moufang loops have isomorphic undirected power graphs, must they have isomorphic directed power graphs? It is known that two groups with isomorphic undirected power graphs must have isomorphic directed power graphs. We are able to extend that result to Moufang loops by showing that Moufang loops with isomorphic undirected power graphs have isomorphic directed power graphs. Along the way we investigate a class of Moufang loops which we term the generalized octonion loops and behave similarly to the generalized quaternion groups.