on the commutativity degree in finite moufang loops
Authors
abstract
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.
similar resources
Generators for Finite Simple Moufang Loops
Moufang loops are one of the best-known generalizations of groups. There is only one countable family of nonassociative finite simple Moufang loops, arising from the split octonion algebras. We prove that every member of this family is generated by three elements, using the classical results on generators of unimodular groups.
full textOn Moufang A-loops
In a series of papers from the 1940’s and 1950’s, R.H. Bruck and L.J. Paige developed a provocative line of research detailing the similarities between two important classes of loops: the diassociative A-loops and the Moufang loops ([1]). Though they did not publish any classification theorems, in 1958, Bruck’s colleague, J.M. Osborn, managed to show that diassociative, commutative A-loops are ...
full textGenerators of Nonassociative Simple Moufang Loops over Finite Prime Fields
The first class of nonassociative simple Moufang loops was discovered by L. Paige in 1956 [9], who investigated Zorn’s and Albert’s construction of simple alternative rings. M. Liebeck proved in 1987 [7] that there are no other finite nonassociative simple Moufang loops. We can briefly describe the class as follows: For every finite field F, there is exactly one simple Moufang loop. Recall Zorn...
full textMoufang Loops of Small Order
The main result of this paper is the determination of all nonassociative Moufang loops of orders *31. Combinatorial type methods are used to consider a number of cases which lead to the discovery of 13 loops of the type in question and prove that there can be no others. All of the loops found are isomorphic to all of their loop isotopes, are solvable, and satisfy both Lagrange's theorem and Syl...
full textPseudo-automorphisms and Moufang Loops
An extensive study of Moufang loops is given in [2].1 One defect of that study is that it assumes Moufang's associativity theorem [6], the only published proof of which involves a complicated induction. Using pseudo-automorphisms along with recent methods of Kleinfeld and the author [S], we shall give simple noninductive proofs of three associativity theorems, one of which (Theorem 5.1) general...
full textHalf-isomorphisms of Moufang Loops
We prove that if the squaring map in the factor loop of a Moufang loop Q over its nucleus is surjective, then every half-isomorphism of Q onto a Moufang loop is either an isomorphism or an anti-isomorphism. This generalizes all earlier results in this vein.
full textMy Resources
Save resource for easier access later
Journal title:
international journal of group theoryPublisher: university of isfahan
ISSN 2251-7650
volume
issue Articles in Press 2015
Hosted on Doprax cloud platform doprax.com
copyright © 2015-2023