On the commutativity degree in finite Moufang loops

Document Type : Research Paper


Tabriz Branch, Islamic Azad University


‎The commutativity degree‎, ‎Pr(G)">Pr(G)Pr(G)‎, ‎of a finite group G">GG (i.e‎. ‎the probability that two (randomly chosen) elements of G">GGcommute with respect to its operation)) has been studied well by many authors‎. ‎It is well-known that the best upper bound for Pr(G)">Pr(G)Pr(G) is 58">5858 for a finite non-abelian group G">GG‎. 
 ‎In this paper‎, ‎we will define the same concept for a finite non--abelian Moufang loop M">MM and try to give a best upper bound for Pr(M)">Pr(M)Pr(M)‎. ‎We will prove that for a well-known class of finite Moufang loops‎, ‎named Chein loops‎, ‎and its modifications‎, ‎this best upper bound is 2332">23322332‎. ‎So‎, ‎our conjecture is that for any finite Moufang loop M">MM‎, ‎Pr(M)≤2332">Pr(M)2332Pr(M)≤2332‎. 
‎Also‎, ‎we will obtain some results related to the Pr(M)">Pr(M)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.


Main Subjects

[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.
‎[5] O‎. ‎Chein and A‎. ‎Rajah‎, ‎Possible orders of non--associative Moufang loops‎, Comment‎. ‎Math‎. ‎Univ‎. ‎Carolin., ‎41 (2000) 237-244‎.
‎[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.
‎[14] P‎. ‎Lescot‎, ‎Isoclinism classes and comutativity degree of finite groups‎, ‎J‎. ‎Algebra‎, 177 (1995) 847-869‎.
‎[15] F‎. ‎Leong and A‎. ‎Rajah‎, ‎Moufang loops of odd order $p^a q_1^2...q_n^2r_1...r_m$, J‎. ‎Algebra, 190 ‎(1997) 474-486‎.
‎[16] D‎. ‎MacHale‎, ‎Commutativity in finite rings‎, ‎Amer‎. ‎Math‎. ‎Monthly‎, 83 (1975) 30-32‎.
‎[17]G‎. ‎P‎. ‎Nagy and M‎. ‎Valsecchi‎, ‎On nilpotent Moufang loops with central associators‎, J‎. ‎Algebra‎, 307 (2007) 547-567‎.
‎[18] G‎. ‎P‎. ‎Nagy and P‎. ‎Vojtvechovsky‎, ‎The Moufang loops of order 64 and 81‎, J‎. ‎Symbolic Comput.‎, ‎42 (2007) 871-883‎.
‎[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.
[20] H‎. ‎O‎. ‎Pflugfelder‎, Quasigroups and loops‎: ‎Introduction, ‎Heldermann Verlag‎, ‎Berlin‎, ‎1990‎.
‎[21] M‎. ‎C‎. ‎Slattery and A‎. ‎L‎. ‎Zenisek‎, ‎Moufang loops of order 243‎, ‎Comment‎. ‎Math‎. ‎Univ‎. ‎Carolin.‎, 53 (2012) 423-428‎.