On the commutativity degree in finite Moufang loops

Document Type : Research Paper


Tabriz Branch, Islamic Azad University


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.


Main Subjects

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