Isoclinism in Moufang loops and characterization of finite simple Moufang loops by non-commuting graph and isoclinism

The non-commuting graph of a non-abelian finite group has received some attention in existing literature. The order of groups in some classes of finite groups have been characterized by their non-commuting graphs. However, it has been proved recently that a finite simple group can be characterized by its non-commuting graph. We have already generalized the two notions, commutativity degree and non-commuting graph for a finite Moufang loop M and tried to characterize some finite non-commutative Moufang loops with their non-commuting graph. Also, it has been proved by Lescot that two isoclinic finite groups have the same commutativity degrees. In this talk, we want to generalize the notion of isoclinism to Moufang loops and show that two isoclinic finite Moufang loops have the same commutativity degrees. Then, we show that the finite Moufang loops with the same commutativity degrees and isomprphic non-commuting graphs are order characterictic and finally, chracterize all finite simple Moufang loops under isomorphism of their non-commuting graphs and isoclinism