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


[1] A. Abdollahi, S. Akbari and H.R. Maimani, Non-commuting graph of a group, J. Algebra, Vol. 298 (2006), pp. 468-492.
[2] K. Ahmadidelir, C.M. Campbell and H. Doostie, Almost Commutative Semigroups, Algebra Colloq., Vol. 18 (Spec 1) (2011), pp. 881-888.
[3] K. Ahmadidelir, On the Commutatively degree in nite Moufang loops, Int. J. Group Theory, Vol. 5 (2016), pp. 37-47.
[4] K. Ahmadidelir, On the non-commuting graph in nite Moufang loops, J. Algebra Appl., Vol. 17 No. 4 (2018), pp. 1850070-1-22.
[5] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, American Elsevier publishing Co., Inc., New York, (1977).
[6] R.H. Bruck, A survay of binary systems, Springer-Verlag, Berlin-Heidelberg, (1958).
[7] M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The strong perfect graph theorem, Ann. of Math., Vol. 164 No. 1 (2006), pp. 51-229.
[8] M.R. Darafsheh, Groups with the same non-commuting graph, Discretre Appl. Math., Vol. 157 No. 4 (2009), pp. 833-837.
[9] S. M. Gagola III, Hall's theorem for Moufang loops, J. Algebra, Vol. 323 No. 12 (2010), pp. 3252-3262.
[10] The GAP group, GAP- Groups, Algorithms and Programming, Aachen, St. Andrews Version 4.9.3 (2018),
(http://www.gap-system.org).
[11] W.H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly, Vol. 80 (1973), pp. 1031-1034.
[12] A.N. Grishkov, A.V. Zavarnitsine, Lagranges theorem for Moufang loops, Math. Proc. Cambridge. Philos. Soc., Vol. 139 No. 1 (2005), pp. 41-57.
[13] A.N. Grishkov, A.V. Zavarnitsine, Sylow's theorem for Moufang loops, J. Algebra, Vol. 321 No. 7 (2009), pp. 1813-1825.
[14] P. Lescot, Isoclinism classes and commutativity degree of nite groups, J. Algebra, Vol. 177 (1995), pp. 847-869.
[15] M.W. Liebeck, The classi cation of nite simple Moufang loops, Math. Proc. Cambridge Philos. Soc., Vol. 102 No. 1 (1987), pp. 33-47.
[16] D. MacHale, Commutativity in nite rings, Amer. Math. Monthly, Vol. 83 (1976), pp. 30-32.
[17] A.R. Moghaddamfar, About noncommuting graphs, Siberian Math. J., Vol. 47 No. 5 (2006), pp. 911-914.
[18] A.R. Moghaddamfar, W.J. Shi, W. Zhou, and A.R. Zokayi, On the noncommuting graph associated with a fi nite group, Siberian Math. J., Vol. 46 No. 2 (2005), pp. 325-332.
[19] L. Paige, A Class of Simple Moufang Loops, Proc. Amer. Math. Soc., Vol. 7 No. 3 (1956), pp. 471-482.
[20] H.O. Pugfelder, Quasigroups and loops: Introduction, Helderman Verlag, Berlin, (1990).
[21] R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory, Vol. 16 (2013), pp. 793-824.