有限牟方环的交换度

IF 0.7 Q2 MATHEMATICS

International Journal of Group Theory Pub Date : 2016-09-01 DOI:10.22108/IJGT.2016.8477

K. Ahmadidelir

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引用次数: 2

摘要

有限群G”>GG(即)的交换度，' ' Pr(G) ' ' >Pr(G)Pr(G) ' '， ' '。许多作者已经很好地研究了G的两个(随机选择的)元素相对于其操作的交换概率。众所周知，对于有限非阿贝尔群G ' >GG '， Pr(G) ' b> Pr(G) ' Pr(G) '的最佳上界是58 ' >5858。在本文中，我们将对有限非阿贝尔牟方环定义相同的概念，并尝试给出Pr(M)”>Pr(M)Pr(M)”的最佳上界。我们将证明对于一类著名的有限Moufang环，称为Chein环及其修正，其最佳上界是2332”bb0 23322332”。因此，我们的猜想是，对于任意有限牟方环M ' >MM '， ' Pr(M)≤2332 ' >Pr(M)≤2332Pr(M)≤2332 '。同时，我们将得到一些关于Pr(M)”>Pr(M)Pr(M)”的结果，并提出在群论中关于有限方环中有限群的结构与其交换度之间的关系的类似问题。

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ON THE COMMUTATIVITY DEGREE IN FINITE MOUFANG LOOPS

‎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.

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来源期刊

International Journal of Group Theory MATHEMATICS-

CiteScore

0.90

自引率

0.00%

发文量

审稿时长

30 weeks

期刊介绍： International Journal of Group Theory (IJGT) is an international mathematical journal founded in 2011. IJGT carries original research articles in the field of group theory, a branch of algebra. IJGT aims to reflect the latest developments in group theory and promote international academic exchanges.