A Generalization of Moufang and Steiner Loops

We study a variety of loops, RIF, which arise naturally from considering inner mapping groups, and a somewhat larger variety, ARIF. All Steiner and Moufang loops are RIF, and all flexible C-loops are ARIF. All ARIF loops are diassociative. Received June 6, 2001; accepted in final form November 14, 2002.     

Incidence properties of cosets in loops

We study incidence properties among cosets of infinite loops, with emphasis on well‐structured varieties such as antiautomorphic loops and Bol loops. While cosets in groups are either disjoint or identical, we find that the incidence structure in general loops can be much richer. Every symmetric design, for example, can be realized as a canonical collection of cosets of a infinite loop. We show that in the variety of antiautomorphic loops the poset formed by set inclusion among intersections of left cosets is isomorphic to that formed by right cosets. We present an algorithm that, given a infinite Bol loop S, can in some cases determine whether |S| divides |Q| for all infinite Bol loops Q with S?Q, and even whether there is a selection of left cosets of S that partitions Q. This method results in a positive confirmation of Lagrange's Theorem for Bol loops for a few new cases of subloops. Finally, we show that in a left automorphic Moufang loop Q (in particular, in a commutative Moufang loop Q), two left cosets of S?Qare either disjoint or they intersect in a set whose cardinality equals that of some subloop of S.     

Quelques consequences de la locale nilpotence des boucles de moufang commutatives

It is well-known that any finitely generated commutative Moufang loop (CML) is centrally nilpotent and has a finite derived subloop. Consequently such a loop possesses all the classical properties of noethe-rianity: any subloop is finitely generated too, any surjective endomorphism is an automorphism, etc. Besides we prove that, in any CML E(finitely generated or not) the maximal subloops are normal of prime index ; thus the Frattini quotient E_/Φ(E) is an abelian group, sub-direct product of groups of prime order. We shall study also some dual notion of the Frattini subloop, namely the subloop φ^*(E) generated by the minimal normal subloops ; it turns out that φ^*(E) is made up by the products of the prime order central elements.     

Semiautomorphic inverse property loops

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 a² is a Moufang element, a³ is a c-element and a⁶ 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.     

Shortest single axioms for commutative moufang loops of exponent 3

In this note, we attempt to find all shortest single product axioms for commutative Moufang loops of exponent 3. These investigations were aided by the automated theorem-prover Prover9 and the model-generator Mace4.     

Zorn's Matrices and Finite Index Subloops#

Zorn's Algebra ?(R) has a multiplicative function called determinant with properties similar to the usual one. The set of elements in ?(R) with determinant 1 is a Moufang loop that we will denote by IΓ. In our main result we prove that if R is a Dedekind algebraic number domain that contains a unit of infinite order, each finite index subloop ?, such that IΓ has the weak Lagrange property relative to ?, is a congruence subloop.     

Structure of finitely generated commutative alternative algebras and special Moufang loops

It is proved that the commutator ideal of the multiplication algebra of a free commutative alternative algebra of rank n is nilpotent of index n ? 1. As a corollary to this fact, the Bruck theorem for special commutative Moufang loops is derived.     

Every diassociative A-loop is Moufang

An A-loop is a loop in which every inner mapping is an automorphism. A problem which had been open since 1956 is settled by showing that every diassociative A-loop is Moufang.

     

Abelian-by-Cyclic Moufang Loops

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 [10 Rajah , A. ( 2001 ). Moufang loops of odd order pq ³ . J. Algebra 235 ( 1 ): 66 – 93 .[Crossref], [Web of Science ®] , [Google Scholar]] loops of this type of odd order 3q ³, with prime q ≡ 1 (mod 3), are particular cases of our series. Some of the examples are shown to be embeddable into a Cayley algebra.     

On the Bruck-Slaby theorem for commutative Moufang loops

With the help of the relationship between commutative Moufang loops and alternative commutative algebras, we prove, rather easily, the following weakened version of the Bruck-Slaby theorem: a finitely generated commutative Moufang loop is centrally nilpotent. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 275–281, August, 1999.     

On Centrally and Nuclearly Nilpotent Moufang Loops

Let Q be a finite Moufang loop with nucleus N(Q) and associator subloop 𝒜(Q). First we prove if the factor loop over the nucleus Q/N has nontrivial center, then the center of Q is nontrivial too. By using this result we prove that the centrally nilpotence of Q/N(Q) implies the centrally nilpotence of 𝒜(Q), and we show that, for the centrally nilpotence of a finite Moufang loop, the centrally nilpotence of Q/N(Q) and Q/𝒜(Q) is a necessary and sufficient condition. Finally, as a corollary we give a necessary and sufficient condition for the equivalence of centrally and nuclearly nilpotence of finite Moufang loops, namely, the centrally nilpotence of Q/𝒜(Q).     

Conjugacy of Sylow 2-Subloops of the Chein Loops M 2n (G, 2)

In general, Sylow's Theorems do not hold for finite Moufang loops. It can be seen that if p is an odd prime then the Sylow p-subloops of the Chein loop M _2n (G, 2) are conjugate. Here we prove that it is also true that all the Sylow 2-subloops of M _2n (G, 2) are conjugate.     

Commutative Moufang Loops with Finite Conjugacy Classes of Subloops

It is proved that the following conditions are equivalent for an arbitrary commutative Moufang loop :1) the loop is finite over the center;2) every subloop of defines a finite conjugacy class of subloops;3) every associative subloop of defines a finite conjugacy class of subloops;4) every infinite associative subloop of defines a finite conjugacy class of subloops.     

On the lattice formed by all normal subloops of a finite loop

All normal subloops of a loopG form a modular latticeL _n (G). It is shown that a finite loopG has a complemented normal subloop lattice if and only ifG is a direct product of simple subloops. In particular,L _n (G) is a Boolean algebra if and only if no two isomorphic factors occurring in a decomposition ofG are abelian groups. The normal subloop lattice of a finite loop is a projective geometry if and only ifG is an elementary abelianp-group for some primep.     

Diassociativity in Conjugacy Closed Loops

Abstract

Let Q be a conjugacy closed loop, and N(Q) its nucleus. Then Z(N(Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is power-associative and |Q| is finite and relatively prime to 6, then Q is a group. If Q is a finite non-associative extra loop, then 16 ∣ |Q|.     

Linear representations of formal loops

A representation of an object in a category is an abelian group in the corresponding comma category. In this paper, we derive the formulas describing linear representations of objects in the category of formal loops and formal loop homomorphisms and apply them to obtain a new approach to the representation theory of formal Moufang loops and Malcev algebras based on Moufang elements. Certain ‘non-associative Moufang symmetry’ of groups is revealed.     

Half-isomorphisms of Finite Automorphic Moufang Loops

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.     

When are inner mapping groups generated by conjugation maps?

A subloop of a loop Q is said to be normal if it is stabilized by all maps in the inner mapping group of Q. Here we show that in many cases, the inner mapping group of a Moufang loop is actually generated by conjugation maps. This includes any Moufang loop whose cubes generate either the entire loop or a subloop of index three. Such a result can be an extremely useful tool when proving that certain subloops are indeed normal just by showing that they are stabilized by the conjugation maps.