Abstract
We prove that for positive integers
$m \geq 1, n \geq 1$
and a prime number
$p \neq 2,3$
there are finitely many finite m-generated Moufang loops of exponent
$p^n$
.
For a given element $g$ of a finite group $G$, the probablility that the commutator of randomly choosen pair elements in $G$ equals $g$ is the relative commutativity degree of $g$. In this paper we are interested in studying the relative commutativity degree of the Dihedral group of order $2n$ and the Quaternion group of order $2^{n}$ for any $n\geq 3$ and we examine the relative commutativity degree of infinite class of the Moufang Loops of Chein type, $M(G,2)$.
Abstract
We construct two infinite series of Moufang loops of exponent 3 whose commutative centre (i. e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of orders 38 and 311 one of which can be defined as the Moufang triplication of the free Burnside group B(3, 3).
In this paper, we produce the product rules of nonassociative Moufang loops of order 81 by using an analytical approach. We then explore all possible presentations on a suitable set of generators, thereby obtaining a total of five nonisomorphic cases. The result is in agreement with the classification by Nagy and Vojtěchovský using the GAP package LOOPS .
The restricted Burnside problem for Moufang loops
