ON THE COMMUTATIVITY DEGREE IN FINITE MOUFANG LOOPS

Abstract

‎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.