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On the commutativity degree in finite Moufang loops | ||
| International Journal of Group Theory | ||
| مقاله 3، دوره 5، شماره 3، آذر 2016، صفحه 37-47 اصل مقاله (211.55 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2016.8477 | ||
| نویسنده | ||
| Karim Ahmadidelir* | ||
| Tabriz Branch, Islamic Azad University | ||
| چکیده | ||
| The \textit{commutativity degree}, $Pr(G)$, of a finite group $G$ (i.e. the probability that two (randomly chosen) elements of $G$ commute with respect to its operation)) has been studied well by many authors. It is well-known that the best upper bound for $Pr(G)$ is $\frac{5}{8}$ for a finite non--abelian group $G$. In this paper, we will define the same concept for a finite non--abelian \textit{Moufang loop} $M$ and try to give a best upper bound for $Pr(M)$. We will prove that for a well-known class of finite Moufang loops, named \textit{Chein loops}, and its modifications, this best upper bound is $\frac{23}{32}$. So, our conjecture is that for any finite Moufang loop $M$, $Pr(M)\leq \frac{23}{32}$. Also, we will obtain some results related to the $Pr(M)$ and ask the similar questions raised and answered in group theory about the relations between the structure of a finite group and its commutativity degree in finite Moufang loops. | ||
| کلیدواژهها | ||
| Loop theory؛ Finite Moufang loops؛ Commutativity degree in finite groups | ||
| مراجع | ||
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[1] K. Ahmadidelir, C. M. Campbell and H. Doostie, Almost Commutative Semigroups, Algebra Colloq., 18 (2011) 881-888. [2] R. H. Bruck, A survey of binary systems, Springer Verlag, Berlin-Göttingen-Heidelberg, 1958. [3] O. Chein, Moufang loops of small order. I, Trans. Amer. Math. Soc., 188 (1974) 31-51. [4] O. Chein, Moufang loops of small order, Mem. Amer. Math. Soc., 13 (1978) iv+131 pp. [6] A. Drapal, How far apart can the group multiplication tables be?, European J. Combin., 13 (1992) 335-343. [7] A. Drapal and P. Vojtvechovsky, Moufang loops that share associator and three quarters of their mulyiplication tables, Rocky Mountain J. Math., 36 (2006) 425-455. [8] S. M. Gagola III, Hall's theorem for Moufang loops, J. Algebra, 323, no. 12 (2010) 3252-3262. [9] The GAP group, emphGAP- Groups, Algorithms and Programming, Aachen, St. Andrews Version 4.7.2, (2013), (http://www.gap--system.org). [10] E. G. Goodaire, S. May and M. Raman, The Moufang Loops of Order Less Than $64$}, Nova Science Publishers, Inc., Commack, NY, 1999. [11] W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly, 80 (1973) 1031-1034. [12] A. N. Grishkov and A. V. Zavarnitsine, Lagrange's theorem for Moufang loops, Math. Proc. Cambridge Philos. Soc., 139, (2005) 41-57. [13] A. N. Grishkov and A. V. Zavarnitsine, Sylow's theorem for Moufang loops, J. Algebra, 321 (2009) 1813-1825. [19] G. P. Nagy and P. Vojtvechovsky, LOOPS Version 2.1.3, Package for GAP 4.4.12., Available at http://www.math.du.edu/loops. | ||
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