Commutativity degree and non-commuting graph in finite groups and Mofang Loops and their relationships

Document Type : Research Paper

Authors

1 Department of Mathematics, Science and Research Branch, Islamic Azad University, P.O. Box 14515-1775, Tehran, Iran.

2 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.

3 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

10.29252/as.2020.1719

Abstract

Terms like commutativity degree, non-commuting graph and isoclinism are far well-known for much of the group theorists nowadays. There are so many papers about each of these concepts and also about their relationships in finite groups. Also, there are some recent researches about generalizing these notions in finite rings and their connexions.
The concepts of commutativity degree and non-commuting graph are also extended to non-associative structures such as Moufang loops and some part of the known results in group theory in these contexts have been expanded to them.
In this paper, we are going to generalize the notion of isoclinism in finite Moufang loops and then study the relationships between these three concepts. Among other results, we prove that two isoclinic finite Moufang loops have the same commutativity degree and if they have the same sizes of centers and commutants then they have isomorphic non-commuting graphs. Also, the converses of these results have been investigated.
Furthermore, it has been proved that a finite simple group can be characterized by its non-commuting graph. We will prove the same is true for a finite simple Moufang loop by imposing one additional hypothesis, namely, the isoclinism of the regarding loops.

Keywords

References

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