LOOPS AND GYRO-GROUPS

Abstract

Graphs provide a natural framework for representing algebraic structures. Motivated by this perspective, we undertake a study of the G-graphs associated with certain finite loops, which generalize the notion of groups. In particular, we consider three distinguished classes of loops: Bol loops, Bruck loops, and Moufang loops and investigate their connections with gyro-groups. For each class, we construct the corresponding G-graphs and examine their structural properties. Moreover, we analyze the G-graphs of selected gyro-groups and compare their characteristics with those of the aforementioned loops. This work offers a loop-theoretic approach to the study of gyro-groups, enriching the understanding of their underlying algebraic structures.