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We prove that a normal subloop X of a Moufang loop Q induces an abelian congruence of Q if and only if u (x y) = (u y) x for all x , y ∈ X and u ∈ Q . This characterization is then used to show that classically solvable finite 3-divisible Moufang loops are congruence solvable. [ABSTRACT FROM AUTHOR]

We initiate a systematic study of abelian-by-cyclic Moufang loops, that is, Moufang loops Q with an abelian normal subgroup X such that Q / X is a cyclic group. Among other results, we construct all split abelian-by-cyclic Moufang loops in which both X and Q / X are 3-divisible, using so-called Moufang permutations on X, which are permutations that deviate from an automorphism of X by an alternating biadditive mapping. Additional abelian-by-cyclic Moufang loops are obtained from so-called construction pairs. As an aside, we show that, in a Moufang loop Q obtained from a construction pair, the abelian normal subgroup X induces an abelian congruence of Q if and only if Q is a group. [ABSTRACT FROM AUTHOR]

The main result is an elementary proof of the nilpotency of Moufang loops which are of prime power order. Besides basic properties of Moufang loops and Sylow theorems, the proof relies on the fact that a relative multiplication group of a (centrally nilpotent) subloop of order p k is a p -group in any finite Moufang loop Q , and that L φ (x) − 1 φ L x φ − 1 and R φ (x) − 1 φ R x φ − 1 are pseudoautomorphisms of Q whenever x ∈ Q and φ is a pseudoautomorphism of Q. [ABSTRACT FROM AUTHOR]

In groups, an abelian normal subgroup induces an abelian congruence. We construct a class of centrally nilpotent Moufang loops containing an abelian normal subloop that does not induce an abelian congruence. On the other hand, we prove that in 6-divisible Moufang loops, every abelian normal subloop induces an abelian congruence. In loops, congruence solvability adopted from the universal-algebraic commutator theory of congruence modular varieties is strictly stronger than classical solvability adopted from group theory. It is an open problem whether the two notions of solvability coincide in Moufang loops. We prove that they coincide in 6-divisible Moufang loops and in Moufang loops of odd order. In fact, we show that every Moufang loop of odd order is congruence solvable, thus strengthening Glauberman's Odd Order Theorem for Moufang loops. [ABSTRACT FROM AUTHOR]

In 1978, Doro, in his paper [S. Doro, Simple moufang loops, Math. Proc. Camb. Philos. Soc. 83 (1978) 377–392] published the following conjecture: If the nucleus of a Moufang loop is trivial, then the commutant is a normal subloop. By working in the multiplication group of the loop we prove that in case of finite Moufang loops with trivial nucleus, the commutant is normal if and only if it is trivial. [ABSTRACT FROM AUTHOR]

One of the earliest examples of public key cryptography is the Diffie-Hellman key exchange protocol, which is based on the discrete logarithm problem. In 2000, Ko, Lee et al. modified the Diffie-Hellman protocol by using braid groups as platform where the security of the protocol lies on the conjugacy search problem. In this paper, we propose two key exchange protocols that use two well known classes of nonassociative Moufang loops as potential platforms. [ABSTRACT FROM AUTHOR]

We shall show that there exist sofic groups which are not locally embeddable into finite Moufang loops. These groups serve as counterexamples to a problem and two conjectures formulated in the paper by M. Vodička, P. Zlatoš (2019).

This paper initiates a unified module theory for four varieties of quasigroups: semisymmetric, semisymmetric idempotent, totally symmetric, and totally symmetric idempotent. These classes correspond, respectively, to extended Mendelsohn triple systems, Mendelsohn triple systems (MTS), extended Steiner triple systems, and Steiner triple systems (STS), which, in turn, correspond to partitions of complete (directed) graphs. Letting V stand for any of the aforementioned quasigroup categories, we determine a ring, Z V Q , such that abelian groups in the slice category V / Q are equivalent to (right) Z V Q -modules. This ring is a quotient of the group algebra of the so-called universal stabilizer of the V -quasigroup Q. We prove that the universal stabilizer of Q in V is the fundamental group of either the graph from which the triple system sources its blocks, or a closely related space. In each of the four varieties, we provide a free product description of Z V Q , and show that the factorizations are indexed by blocks of the triple system. We also consider the ranks of free group rings that appear in the coproduct for Z V Q (for the cases V = MTS , STS ), and we describe these ranks in terms of pentagonal numbers. We establish a relationship between the module theory of MTS and commutative Moufang loops of exponent 3. As an application, we show how the MTS module theory completely accounts for the distributive, nonmedial quasigroups of order 81. [ABSTRACT FROM AUTHOR]

Glauberman [10] and Doro [8] studied the structure of Moufang loops of odd order. Glauberman [10] proved that Feit–Thompson's Theorem can be extended to Moufang loops, namely every Moufang loop of odd order is solvable. It turned out that the multiplication group G of a Moufang loop of odd order with trivial nucleus is a group with triality, i.e. S 3 ≤ Aut G with special identities. One of the main problems in Moufang loops: Does there exist a Moufang loop of odd order with trivial nucleus? We give a negative answer by proving that every Moufang loop of odd order has nontrivial nucleus. [ABSTRACT FROM AUTHOR]

In this work, we discovered a dozen of new loop identities we called identities of 'second Bol-Moufang type'. This was achieved by using a generalized and modified nuclear identification model originally introduced by Drápal and Jedlička. Among these twelve identities, eight of them were found to be distinct (from well known loop identities), among which two pairs axiomatize the weak inverse property power associative conjugacy closed (WIP PACC) loop. The four other new loop identities individually characterize the Moufang identities in loops. Thus, now we have eight loop identities that characterize Moufang loops. We also discovered two (equivalent) identities that describe two varieties of Buchsteiner loops. In all, only the extra identities which the Drápal and Jedlička nuclear identification model tracked down could not be tracked down by our own nuclear identification model. The dozen laws ... induced by our nuclear identification form four cycles in the following sequential format: ... and also form six pairs of dual identities. With the help of twisted nuclear identification, we discovered six identities of lengths five that describe the abelian group variety and commutative Moufang loop variety (in each case). The second dozen identities ... nduced by our twisted nuclear identification were also found to form six pairs of dual identities. Some examples of loops of smallest order that obey non-Moufang laws (which do not necessarily imply the other) among the dozen laws ... were found. [ABSTRACT FROM AUTHOR]

We shall show that there exist sofic groups which are not locally embeddable into finite Moufang loops. These groups serve as counterexamples to a problem and two conjectures formulated in the paper by M.Vodička, P. Zlatoš (2019). [ABSTRACT FROM AUTHOR]

In this paper, we study some properties of the non-commuting graph Γ M of a finite Moufang loop M, a graph obtained by setting all non-central elements of M as the vertex set and defining two distinct vertices to be adjacent if and only if their commutator is non-identity. In particular, Hamiltonian as well as (weak) perfectness of non-commuting graphs of Chein loops are considered. We find several upper and lower bounds for commutativity degree of some classes of finite Moufang loops by means of edge number of their non-commuting graphs and algebraic properties. [ABSTRACT FROM AUTHOR]

A loop (Q, ∙) is called a FRUTE loop if it obeys the identity (x ∙ xy)z = (y ∙ zx)x. Interestingly, a FRUTE loop is a Moufang loop, but not neccessarily an extra loop or a group (and vice versa). In this paper, some new algebraic properties of a FRUTE loop are established. It is shown that a FRUTE loop is centrum square, a middle A-Ioop and obeys the law x-1yx = xyx-1. It is proved that in a FRUTE loop, the following loops are equivalent: central square loop, group, extra loop, LC-Ioop, RC-Ioop, C-loop, LNS-loop, RNS-loop, MNS-loop, left Cheban loop, right Cheban loop, Cheban loop, LCC-loop, RCC-Ioop and CC-Ioop. [ABSTRACT FROM AUTHOR]

The structure of finite quasi-fields with associative degrees is investigated. These are, above all, associative quasifields, called near-fields. These also include the Moufang quasifields which have loops of nonzero elements are, by definition, loops introduced by Ruth Moufang in 1935. The paper presents the main definitions associated with quasifields. It is shown that identity element of any finite (right) quasifield $Q$ generates a simple subfield $P$, and $Q$ is always a one-sided module over $P$ , and a two-way~--- is not always. As a result, found new proof of a well-known statement: a simple subfield of a finite semifield always lies in the center. At the same time, the finite near-fields with a simple subfield that does not lie in the center. Famous the question of maximal subfields of finite quasifields is completely solved for a class of finite near-fields of order $p^r$ with prime numbers $p$ and $r$. In solving the questions about maximal subfields and spectra of group orders of nonzero elements of finite Moufang quasifields, it is proposed to use the well-known analogues of the group-theoretic theorems of Lagrange and Sylow. All possible two-digit orders of the proper Moufang quasifields are listed.

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 2n for any n ≥ 3 and we examine the relative commutativity degree of infinite class of the Moufang Loops of Chein type, M(G, 2). [ABSTRACT FROM AUTHOR]

This is a survey of results from the rich theory of medial and more general quasigroups and its close connection with the geometry of point-reflections. The emphasis is on questions regarding the simplest axiomatization, in the sense of the minimal number of variables appearing in the identtties, on dependence or independence of axioms, and on representation theorems in the style of Toyoda. [ABSTRACT FROM AUTHOR]

The question "Are the loops with universal (i.e. invariant under the isotopy of loops) flexibility law xy ·x = x·yx, middle Bol loops?" is open in the theory of loops. If this conjecture is true then the loops for which all isostrophic loops are flexible are Moufang loops. In the present paper we prove that commutative loops with invariant flexibility under the isostrophy of loops are Moufang loops. In particular, we obtain that commutative IP-loops with universal flexibility are Moufang loops. [ABSTRACT FROM AUTHOR]

The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer n ≥ 2 , the nth-commutativity degree of a finite algebraic structure S, denoted by P n (S) , is the probability that for chosen randomly two elements x and y of S, the relator x n y = y x n holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the nth-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for nth-commutativity degree of these loops, we will obtain best upper bounds for this probability. [ABSTRACT FROM AUTHOR]

Using isotopical approach we generalize concept of binary Moufang loop on n-ary (n > 2) case. We give two examples of ternary Moufang loop which are not a ternary group. [ABSTRACT FROM AUTHOR]

It has been proven that all Moufang loops of order p4q3 where p and q are odd primes with 3 are associative if and only if q≢1(modp). Also all Moufang loops of order p13p23⋯pn3 where each pi are a prime with 3≤p1 were proven to be associative if and only if pi≢1(modpj). In this paper, we extend these results to Moufang loops of order p4q13q23⋯qn3 where p and each qi is a prime with 3 and qi≢1(modqj). [ABSTRACT FROM AUTHOR]

It is known that all Moufang loops of order $$p^4$$ are associative if p is a prime greater than 3. Also, nonassociative Moufang loops of order $$p^5$$ (for all primes p) and $$pq^3$$ (for distinct odd primes p and q, with the necessary and sufficient condition $$q\equiv 1({\text{ mod }}\ p)$$ ) have been proved to exist. Consider a Moufang loop L of order $$p^{\alpha }q^{\beta }$$ where p and q are odd primes with $$p3$$ , $$\alpha \le 4$$ and $$\beta \le 3$$ . [ABSTRACT FROM AUTHOR]

The first paper in this series initiated a study of Sylow theory for quasigroups and Latin squares based on orbits of the left multiplication group. The current paper is based on so-called pseudo-orbits, which are formed by the images of a subset under the set of left translations. The two approaches agree for groups, but differ in the general case. Subsets are described as sectional if the pseudo-orbit that they generate actually partitions the quasigroup. Sectional subsets are especially well behaved in the newly identified class of conflatable quasigroups, which provides a unified treatment of Moufang, Bol, and conjugacy closure properties. Relationships between sectional and Lagrangean properties of subquasigroups are established. Structural implications of sectional properties in loops are investigated, and divisors of the order of a finite quasigroup are classified according to the behavior of sectional subsets and pseudo-orbits. An upper bound is given on the size of a pseudo-orbit. Various interactions of the Sylow theory with design theory are discussed. In particular, it is shown how Sylow theory yields readily computable isomorphism invariants with the resolving power to distinguish each of the 80 Steiner triple systems of order 15. [ABSTRACT FROM AUTHOR]

Code loops are certain Moufang 2-loops constructed from doubly even binary codes that play an important role in the construction of local subgroups of sporadic groups. More precisely, code loops are central extensions of the group of order 2 by an elementary abelian 2-group V in the variety of loops such that their squaring map, commutator map and associator map are related by combinatorial polarization and the associator map is a trilinear alternating form. Using existing classifications of trilinear alternating forms over the field of 2 elements, we enumerate code loops of dimension d = dim ⁡ ( V ) ≤ 8 (equivalently, of order 2 d + 1 ≤ 512 ) up to isomorphism. There are 767 code loops of order 128, and 80826 of order 256, and 937791557 of order 512. [ABSTRACT FROM AUTHOR]

A Moufang loop M of order 3 is constructed, together with a pair a, b of elements of M, such that the set of all elements of M associating with a and b is not a subloop. This also gives an example of a nonassociative Moufang loop with a generating set in which every three elements have trivial associator. [ABSTRACT FROM AUTHOR]

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 -loops for some primes , 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. [ABSTRACT FROM AUTHOR]

Quantum quasigroups are algebraic structures providing a general self-dual framework for the nonassociative extension of Hopf algebra techniques. They also have one-sided analogues, which are not self-dual. The paper presents a survey of recent work on these structures, showing how they furnish various solutions to the quantum Yang--Baxter equation. [ABSTRACT FROM AUTHOR]

We show that each half-automorphism of a finite automorphic Moufang loop is trivial. In general, this is not true for finite left automorphic Moufang loops and for finite automorphic loops. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]