On finite loops with nilpotent inner mapping groups

We shall show how the nilpotency class of a finite loop Q is determined by the properties of a nilpotent inner mapping group. We also show that a classical result by Baer on the structure of abelian finite capable groups holds for Moufang loops of odd order.     

On nilpotent Moufang loops with central associators

In this paper, we investigate Moufang p-loops of nilpotency class at least three for $p>3$. The smallest examples have order $p^5$ and satisfy the following properties: (1) They are of maximal nilpotency class, (2) their associators lie in the center, and (3) they can be constructed using a general form of the semidirect product of a cyclic group and a group of maximal class. We present some results concerning loops with these properties. As an application, we classify proper Moufang loops of order $p^5$, $p>3$, and collect information on their multiplication groups.     

Primary Decompositions in Varieties of Commutative Diassociative Loops

The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially satisfied in the abelian group case) that all nth powers are central, for a fixed n. For n = 2, we get precisely commutative C loops. For n = 3, a prominent variety is that of commutative Moufang loops.

Many analogies between commutative C and Moufang loops have been noted in the literature, often obtained by interchanging the role of the primes 2 and 3. We show that the correct encompassing variety for these two classes of loops is the variety of commutative RIF loops. In particular, when Q is a commutative RIF loop: all squares in Q are Moufang elements, all cubes are C elements, Moufang elements of Q form a normal subloop M ₀(Q) such that Q/M ₀(Q) is a C loop of exponent 2 (a Steiner loop), C elements of L form a normal subloop C ₀(Q) such that Q/C ₀(Q) is a Moufang loop of exponent 3. Since squares (resp., cubes) are central in commutative C (resp., Moufang) loops, it follows that Q modulo its center is of exponent 6. Returning to the decomposition theorem, we find that every torsion, commutative RIF loop is a direct product of a C 2-loop, a Moufang 3-loop, and an abelian group with each element of order prime to 6.

We also discuss the definition of Moufang elements and the quasigroups associated with commutative RIF loops.     

On connected transversals to abelian subgroups and loop theoretical consequences

In this paper one of our questions is the following: Which finite abelian groups are (are not) isomorphic to inner mapping groups of loops? It is well known that if the inner mapping group of a finite loop Q is abelian, then Q is centrally nilpotent. The other question is: Which properties of abelian inner mapping groups imply the central nilpotency of class at most two of the loop? After reminding the reader of the known results we show new ones. To solve these problems we transform them into group theoretical problems, then using connected transversals we get some answer. Received: 1 December 2004; revised: 8 November 2005     

Moufang sets from groups of mixed type

We consider the existence of Moufang sets related to certain groups of mixed type. This way, we obtain new examples of Moufang sets and new constructions of known classes. The most interesting class of new examples is related to the Moufang quadrangles of type and to the Ree–Tits octagon over a nonperfect field, and the root groups of each member have nilpotency class three.     

Left conjugacy closed loops of nilpotency class two

Every LCC loop Q with Inn Q abelian is nilpotent class two. A loop Q of nilpotency class two is LCC ? L(x, y) = L(y, x) for all x, y ∈ Q ? ?/Z(Mlt Q) is abelian ? [x, y, z] = [x,z,y] for all x, y, z ∈ Q ? [x, y, z] = [xy, z][x, z]^?1 for all x, y, z ∈ Q. All nilpotent LCC loops of order p² are described, and some of their multiplication groups are computed.     

Every Moufang loop of odd order with nontrivial commutant has nontrivial center

We get a partial result for Phillips’ problem: does there exist a Moufang loop of odd order with trivial nucleus? First we show that a Moufang loop Q of odd order with nontrivial commutant has nontrivial nucleus, then, by using this result, we prove that the existence of a nontrivial commutant implies the existence of a nontrivial center in Q. Introducing the notion of commutantly nilpotence, we get that the commutantly nilpotence is equivalent to the centrally nilpotence for the Moufang loops of odd order.     

Constructions of Commutative Automorphic Loops

A loop whose inner mappings are automorphisms is an automorphic loop (or A-loop). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of 2. We initiate the classification of commutative A-loops of small orders and also of order p ³, where p is a prime.     

Coxeter–Chein Loops

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 ?₂ by a variation on a semidirect product. We first settle an open problem, originally proposed by Petr Vojtěchovský in 2003, by finding a minimal presentation for the Chein loop with respect to a presentation for W. We then 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. In particular, even if the Moufang loop is infinite, it is finitely presented. Viewing these presentations as amalgams of loops, we then apply methods due to Blok and Hoffman to describe a family of twisted Coxeter–Chein loops.     

Loops whose loop rings in characteristic 2 are alternative

In this paper, the authors continue their investigation of loops which give rise to alternative loop rings. If the coefficient ring has characteristic 2, these loops turn out to form a surprisingly wide class, in contrast to the situation of characteristic ≠ 2. This paper describes many properties of this class, includes diverse examples of Moufang loops which are united by the fact that they have loop rings which are alternative, and discusses analogues in loop theory of a number of important group theoretic constructions.     

Simple Bol Loops

In this article we investigate the Bol loops and connected with them groups. We prove an analog of the Doro's theorem for Moufang loops and find a criterion for simplicity of Bol loops. One of the main results obtained is the following: If the right multiplication group of a connected finite Bol loop S is a simple group, then S is a Moufang loop.     

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.     

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.     

The smallest moufang loop revisited

We derive presentations for Moufang loops of type M _2n(G, 2), defined by Chein, with G finite, two-generated. We then use G = S ₃ to visualize the smallest non-associative Moufang loop.     

Groupoids with the identity defining the commutative Moufang loops

It is known that in the class of loops the identity xy · xz = x ² · yz defines the commutative Moufang loops. In the article, we consider groupoids with this identity and establish properties of some classes of such groupoids.     

Subloops of indecomposable RA loops

Summary An RA loop is a (necessarily Moufang) loop whose loop rings in any characteristic are alternative, but not associative. There are seven classes of finite indecomposable RA loops. In this paper, we find the indecomposable subloops and the indecomposable nonabelian groups which can appear inside the loops in each class.     

Counterexamples to a rank analog of the Shepherd-Leedham-Green-Mckay theorem on finite p-groups of maximal nilpotency class

By the Shepherd-Leedham-Green-McKay theorem on finite p-groups of maximal nilpotency class, if a finite p-group of order p ⁿ has nilpotency class n?1, then f has a subgroup of nilpotency class at most 2 with index bounded in terms of p. Some counterexamples to a rank analog of this theorem are constructed that give a negative solution to Problem 16.103 in The Kourovka Notebook. Moreover, it is shown that there are no functions r(p) and l(p) such that any finite 2-generator p-group whose all factors of the lower central series, starting from the second, are cyclic would necessarily have a normal subgroup of derived length at most l(p) with quotient of rank at most r(p). The required examples of finite p-groups are constructed as quotients of torsion-free nilpotent groups which are abstract 2-generator subgroups of torsion-free divisible nilpotent groups that are in the Mal’cev correspondence with “truncated” Witt algebras.     

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.     

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.