C-loops are loops satisfying $x(y(yz))=((xy)y)z$. They often behave analogously to Moufang loops and they are closely related to Steiner triple systems and combinatorics. We initiate the study of C-loops by proving: (i) Steiner loops are C-loops, (ii) C-loops are alternative, inverse property loops with squares in the nucleus, (iii) the nucleus of a C-loop is a normal subgroup, (iv) C-loops modulo their nucleus are Steiner loops, (v) C-loops are power associative, power alternative but not necessarily diassociative, (vi) torsion commutative C-loops are products of torsion abelian groups and torsion commutative 2-C-loops; and several other results. We also give examples of the smallest nonassociative C-loops, and explore the analogy between commutative C-loops and commutative Moufang loops.
Andrew Rajah posed at the Loops'11 Conference in Trest, Czech Republic, the following conjecture: Is every variety of loops that satisfies Moufang's theorem contained in the variety of Moufang loops? This paper is motivated by that problem. We give a partial answer to this question and present two types of Steiner loops, one that satisfies Moufang's theorem and another that does not, and neither is Moufang loop.
A loop $X$ is said to satisfy Moufang's theorem if for every $x,y,z\in X$ such that $x(yz)=(xy)z$ the subloop generated by $x$, $y$, $z$ is a group. We prove that the variety $V$ of Steiner loops satisfying the identity $(xz)(((xy)z)(yz)) = ((xz)((xy)z))(yz)$ is not contained in the variety of Moufang loops, yet every loop in $V$ satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah.
We define a variety of loops called semiautomorphic, inverse property loops that generalize Moufang and Steiner loops. We first show an equivalence between a previously studied variety of loops. Next we extend several known results for Moufang and Steiner loops. That is, the commutant is a subloop and if $a$ is in the commutant, then $a^{2}$ is a Moufang element, $a^{3}$ is a $c$-element and $a^{6}$ is in the center. Finally, we give two constructions for semiautomorphic inverse property loops based on Chein's and de Barros and Juriaans' doubling constructions.
An open problem in theory of loops is to find the variety of non- Moufang loops satisfying the Moufang Theorem. In this note, we present a variety of local smooth diassociative loops with such property.
In this study we introduce $α$-elasticity property for generalized Moufang loops. Necessary and sufficient conditions for $α$-elasticity of generalized Moufang loops to be universal are given. Using the universal conditions, and in some cases, with the newly introduced right and left $α$-alternative laws for generalized Moufang loops, some properties of generalized Moufang loops are studied. Condition under which the generalized Moufang loop is an abelian group is stated.
Using groups with triality we obtain some general multiplication formulas in Moufang loops, construct Moufang extensions of abelian groups, and describe the structure of minimal extensions for finite simple Moufang loops over abelian groups.
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.
We investigate the relation between the structure of a Moufang loop and its inner mapping group. Moufang loops of odd order with commuting inner mappings have nilpotency class at most two. $6$-divisible Moufang loops with commuting inner mappings have nilpotency class at most two. There is a Moufang loop of order $2^{14}$ with commuting inner mappings and of nilpotency class three.
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…
We classify Moufang loops of order 64 and 81 up to isomorphism, using a linear algebraic approach to central loop extensions. In addition to the 267 groups of order 64, there are 4262 nonassociative Moufang loops of order 64. In addition to the $15$ groups of order $81$, there are $5$ nonassociative Moufang loops of order $81$, $2$ of which are commutative.
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.
We construct a Moufang loop $M$ of order $3^{19}$ and a pair $a,b$ of its elements such that the set of all elements of $M$ that associate with $a$ and $b$ does not form a subloop. This is also an example of a nonassociative Moufang loop with a generating set whose every three elements associate.
We construct two infinite series of Moufang loops of exponent $3$ whose commutative center (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 $3^8$ and $3^{11}$ one of which can be defined as the Moufang triplication of the free Burnside group $B(3,3)$.
Power graphs of both groups and semigroups have been widely studied. While the power graph of a quasigroup can be defined analogously to that of a group, power graphs of quasigroups and loops have thus far been little studied. In this paper we begin transferring results on the power graphs of groups to the context of loops by addressing a question posed by Peter Cameron: if two Moufang loops have isomorphic undirected power graphs, must they have isomorphic directed power graphs? It is known that two groups with isomorphic undirected power graphs must have isomorphic directed power graphs. We …
We prove that a normal subloop $X$ of a Moufang loop $Q$ induces an abelian congruence of $Q$ if and only if each inner mapping of $Q$ restricts to an automorphism of $X$ and $u(xy) = (uy)x$ for all $x,y\in X$ and $u\in Q$. The former condition can be omitted when $X$ is $3$-divisible. This characterization is then used to show that classically solvable finite $3$-divisible Moufang loops are congruence solvable.
We investigate Moufang loops which can be written as the semidirect product of a loop and a group. We also examine a particular class of loop extensions which arise as a result of a finite cyclic group acting as a group of semiautomorphisms on an inverse property loop. A question posed by Gagola III related to such extensions is partially answered, and we give several examples which illustrate that, in general, these extensions are not well-behaved.
We use groups with triality to construct a series of nonassociative Moufang loops. Certain members of this series contain an abelian normal subloop with the corresponding quotient being a cyclic group. In particular, we give a new series of examples of finite abelian-by-cyclic Moufang loops. The previously known [A. Rajah, J. Alg., 235 (2001), 66-93] loops of this type of odd order 3q^3, with prime q congruent to 1 mod 3, are particular cases of our series. Some of the examples are shown to be embeddable into a Cayley algebra.
We give an interpretation of the construction of torsors from preceding work (Bertram, Kinyon: Associative Geometries. I, J. Lie Theory 20) in terms of classical projective geometry. For the Desarguesian case, this leads to a reformulation of certain results from lot.cit., whereas for the Moufang case the result is new. But even in the Desarguesian case it sheds new light on the relation between the lattice structure and the algebraic structures of a projective space.
It is well known that if a group G factorizes as G = NH where H\leq G and N is normal in G then the group structure of G is determined by the subgroups H and N, the intersection of N with H and how H acts on N with a homomorphism f : H -> Aut(N). Here we generalize the idea by creating extensions using the semi-automorphism group of N. We show that if G=NH is a Moufang loop, N is a normal subloop, and H=<u> is a finite cyclic group of order coprime to three then the binary operation of G depends only on the binary operation of N, the intersection of N with H, and how u permutes the elements of…
Let a Moufang loop Q contain a non-unitary subloop, which is a simple loop. Then Q is not embedded into a loop of invertible elements of any alternative algebra.
We study abelian-by-cyclic Moufang loops. We construct all split $3$-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups $(X,+)$, which are permutations that deviate from an automorphism of $(X,+)$ by an alternating biadditive mapping (satisfying certain properties). More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, we show that in the Moufang loops $Q$ obtained from a construction pair on $(X,+)$ the abelian normal subgroup $(X,+)$ induces an abelian congruence of $Q$ if and only if …
We introduce the concept of propagating equations and focus on the case of associativity propagating in varieties of loops. An equation $\varepsilon$ propagates in an algebra $X$ if $\varepsilon(\overrightarrow y)$ holds whenever $\varepsilon(\overrightarrow x)$ holds and the elements of $\overrightarrow y$ are contained in the subalgebra of $X$ generated by $\overrightarrow x$. If $\varepsilon$ propagates in $X$ then it propagates in all subalgebras and products of $X$ but not necessarily in all homomorphic images of $X$. If $\mathcal V$ is a variety, the propagating core $\mathcal V_{[\varep…
There are a number of identities which, if satisfied by a Bol loop, imply that the loop is actually Moufang. In this paper we show that in a number of cases, the Moufang identity is also forced not by a single identity, but by giving elements a choice of equations to satisfy.
Solving functional equations given in \cite{nagy} for extensions of a group $A$ by a weighted Steiner loop $S$ we obtain concrete description for all loops with interesting weak associativity properties if the Steiner loop $S$ induces only the trivial automorphism on $A$. We show that the restricted Fischer groups and their geometry play an important role for loop extension with right alternative property. Also the automorphism groups of these extensions as well as the conditions for isomorphisms between two extensions are studied.
In this work we construct free Moufang loop in the variety generated by code loops. We apply this construction for study the code loops. Moreover, we define and determine all basic representations of code loops of rank 3 and 4.
The paper is devoted to the study of free objects in the variety of Steiner loops and of the combinatorial structures behind them, focusing on their automorphism groups. We prove that all automorphisms are tame and the automorphism group is not finitely generated if the loop is more than $3$-generated. For the free Steiner loop with $3$ generators we describe the generator elements of the automorphism group and some relations between them.
In 1974 Orin Chein discovered a new family of Moufang loops which are now called Chein loops. Such a loop can be created from any group $W$ together with $\mathbb{Z}_2$ by a variation on a semi-direct product. We study these loops in the case where $W$ is a Coxeter group and show that it has what we call a Chein-Coxeter system, a small set of generators of order 2, together with a set of relations closely related to the Coxeter relations and Chein relations. As a result we are able to give amalgam presentations for Coxeter-Chein loops. This is to our knowledge the first such presentation for a…
Let $Q$ be a loop. If $S\leq Q$ is such that $φ(S) \subseteq S$ for each standard generator of $\mathrm{Inn}(Q)$, then $S$ does not have to be a normal subloop. In an LC loop the left and middle nucleus coincide and form a normal subloop. The identities of Osborn loops are obtained by applying the idea of nuclear identification, and various connections of Osborn loops to Moufang and CC loops are discussed. Every Osborn loop possesses a normal nucleus, and this nucleus coincides with the left, the right and the middle nucleus. Loops that are both Buchsteiner and Osborn are characterized as loop…
The Moufang loop named for Richard Parker is a central extension of the extended binary Golay code. It the prototypical example of a general class of nonassociative structures known today as code loops, which have been studied from a number of different algebraic and combinatorial perspectives. This expository article aims to highlight an experimental approach to computing in code loops, by a combination of a small amount of precomputed information and making use of the rich identities that code loops' twisted cocycles satisfy. As a byproduct we demonstrate that one can reconstruct the multipl…
