ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF FINITE MOUFANG LOOP

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, $M(G,2)$.

New algebraic properties of middle Bol loops II

A loop (Q, ·, \, /) is called a middle Bol loop (MBL) if it obeys the identity x(yz\x)=(x/z)(y\x). To every MBL corresponds a right Bol loop (RBL) and a left Bol loop (LBL). In this paper, some new algebraic properties of a middle Bol loop are established in a different style. Some new methods of constructing a MBL by using a non-abelian group, the holomorph of a right Bol loop and a ring are described. Some equivalent necessary and sufficient conditions for a right (left) Bol loop to be a middle Bol loop are established. A RBL (MBL, LBL, MBL) is shown to be a MBL (RBL, MBL, LBL) if and only if it is a Moufang loop.

Malcev algebras corresponding to smooth almost left automorphic Moufang loops

In this note, we introduce the concept of an almost left automorphic Moufang loop and study the properties of tangent algebras of smooth loops of this class.

On the non-commuting graph in finite Moufang loops

The non-commuting graph associated to a non-abelian group [Formula: see text], [Formula: see text], is a graph with vertex set [Formula: see text] where distinct non-central elements [Formula: see text] and [Formula: see text] of [Formula: see text] are joined by an edge if and only if [Formula: see text]. The non-commuting graph of a non-abelian finite group has received some attention in existing literature. Recently, many authors have studied the non-commuting graph associated to a non-abelian group. In particular, the authors put forward the following conjectures: Conjecture 1. Let [Formula: see text] and [Formula: see text] be two non-abelian finite groups such that [Formula: see text]. Then [Formula: see text]. Conjecture 2 (AAM’s Conjecture). Let [Formula: see text] be a finite non-abelian simple group and [Formula: see text] be a group such that [Formula: see text]. Then [Formula: see text]. Some authors have proved the first conjecture for some classes of groups (specially for all finite simple groups and non-abelian nilpotent groups with irregular isomorphic non-commuting graphs) but in [Moghaddamfar, About noncommuting graphs, Sib. Math. J. 47(5) (2006) 911–914], Moghaddamfar has shown that it is not true in general with some counterexamples to this conjecture. On the other hand, Solomon and Woldar proved the second conjecture, in [R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory 16 (2013) 793–824]. In this paper, we will define the same concept for a finite non-commutative Moufang loop [Formula: see text] and try to characterize some finite non-commutative Moufang loops with their non-commuting graph. Particularly, we obtain examples of finite non-associative Moufang loops and finite associative Moufang loops (groups) of the same order which have isomorphic non-commuting graphs. Also, we will obtain some results related to the non-commuting graph of a finite non-commutative Moufang loop. Finally, we give a conjecture stating that the above result is true for all finite simple Moufang loops.

Variety of loops generated by code loops

In this work, we construct free infinitely generated Moufang loop in the variety generated by code loops and find the minimal set of identities that define this variety. We apply this construction to the study of code loops.

Finite FRUTE loops

A loop [Formula: see text] is called a FRUTE loop if it obeys the identity [Formula: see text]. Interestingly, a FRUTE loop is a Moufang loop but not necessarily an extra loop or a group (and vice versa). In this paper, algebraic properties of the left (right) regular representation set of a FRUTE loop are deduced. A FRUTE loop is shown to be universal and an [Formula: see text]-loop for all [Formula: see text]. A Moufang loop is shown to be a FRUTE loop if and only if it is nuclear cube if and only if it is an [Formula: see text]-loop. It is established that: the smallest, associative, non-commutative FRUTE loop is of order [Formula: see text] (the quaternion group [Formula: see text]); for any [Formula: see text], there exists at least a non-commutative group of order [Formula: see text] that is a FRUTE loop; there exists [Formula: see text]-groups of orders [Formula: see text] that are non-commutative FRUTE loops; there are no non-commutative groups that are FRUTE loops of the following range of orders [Formula: see text]; there are two non-associative FRUTE loops of order [Formula: see text] up to isomorphism and there are six non-isomorphic, non-associative FRUTE loops of order [Formula: see text]. It is noted that there exists a non-associative and non-commutative FRUTE loop of order [Formula: see text]. The study is concluded with some questions, conjectures and problem.

Associative subalgebras of the octonians

In the mid 1970s, Michel Racine classified the maximal subalgebras of an octonian algebra. In this paper, we classify the maximal associative subalgebras. It turns out that there are four, up to isomorphism, all of dimension [Formula: see text]. In final sections, we apply our findings to investigate the groups that sit inside the Moufang loop of invertible elements of the split octonians and also to show that a well-known inequality of Jørgensen holds in a new context.

A characterization of nilpotent Moufang loops of odd order

Glauberman and Wright in [G. G. Glauberman and C. R. B. Wright, Nilpotence of finite Moufang 2-loops, J. Algebra 8 (1968) 415–417] proved that a nilpotent Moufang loop is the direct product of [Formula: see text]-loops for some primes [Formula: see text], consequently the elements of coprime order commute in a nilpotent Moufang loop. In this paper, we prove that in Moufang loops of odd order this condition is equivalent to the central nilpotence.

On Moufang loops that are abelian groups over the nucleus

We give some necessary and sufficient conditions for the equivalency of the following two properties for a Moufang loop Q: Q over the nucleus Q/N is an abelian group and Q over the center is a group. We study those properties of Moufang loops which guarantee that L(y, x) is in the automorphism group of the loop. These imply numerous statements, among them there are well-known old results.