Bol loops as sections in semi-simple Lie groups of small dimension

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most nine-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are isotopic to direct products of Bruck loops of hyperbolic type or to Scheerer extensions of Lie groups by Bruck loops of hyperbolic type.This paper was supported by DAAD.     

On K-Loops Of Finite Order

In this note we undertake an axiomatic investigation of K-loops (or gyrogroups, as A.A.Ungar used to name them) and provide new construction methods for finite K-loops. It is shown how, more or less, the axioms are independent from each other. Especially (K6) is independent as A.A. Ungar already had conjectured. We begin with right loops (L,⊕) and add step by step further properties. So the connection between K-loops, Bol-loops, Bruck-loops and the homogeneous loops of Kik-kawa became clear. The smallest examples of proper K-loops possess 8 elements; there are exactly 3 non-isomorphic of these. At last it is shown that one gets quite naturally a Frobenius-group as a quasidirect product of a K-loop (L,⊕) and a group D of automorphisms of (L,⊕) if D is fixed point free except from 0.     

Beispiele endlicher und unendlicher K-Loops

In this note examples are given for non trivial K-loops. There are commutative examples as well as non commuative, finite examples as well as infinite. Furthermore it will be shown that under an additional condition K-loops and Brück loops coincide.     

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.     

Polar Decomposition of Locally Finite Groups

The study of loops as transversals in groups dates back to the works of Reinhold Baer. In the past few years there have been several papers using polar decomposition in linear algebra in order to construct Bruck loops. In this paper we generalize the notion of polar decomposition to any arbitrary group, and we show that in any polar decomposition the binary operation “inherited” from the group leads to the construction of a Bruck loop.     

A Class of Loops Categorically Isomorphic to Bruck Loops of Odd Order

We define a new variety of loops, Γ-loops. After showing Γ-loops are power-associative, our main goal is showing a categorical isomorphism between Bruck loops of odd order and Γ-loops of odd order. Once this has been established, we can use the well known structure of Bruck loops of odd order to derive the Odd Order, Lagrange and Cauchy Theorems for Γ-loops of odd order, as well as the nontriviality of the center of finite Γ-p-loops (p odd). Finally, we answer a question posed by Jedli?ka, Kinyon and Vojtěchovský about the existence of Hall π-subloops and Sylow p-subloops in commutative automorphic loops.     

Finite Bruck loops

Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops , showing that is essentially the direct product of a Bruck loop of odd order with a -element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite -element Bruck loops are -loops, leaving open the question of whether such obstructions actually exist.

     

Left loops, bipartite graphs with parallelism and bipartite involution sets

We describe a representation of any semiregularleft loop by means of asemiregular bipartite involution set or, equivalently, a 1-factorization (i.e., a parallelism) of a bipartite graph, with at least one transitive vertex. In these correspondences,Bol loops are associated on one hand toinvariant regular bipartite involution sets and, on the other hand, totrapezium complete bipartite graphs with parallelism; K-loops (or Bruck loops) are further characterized by a sort of local Pascal configuration in the related graph. Research partially supported by the Research Project of M.I.U.R. (Italian Ministry of Education, University and Research) “Strutture geometriche, combinatoria e loro applicazioni” and by the Research group G.N.S.A.G.A. of INDAM.     

Nakayama Automorphisms of Generalized Co-Frobenius Algebras

Bruck loops with abelian inner mapping groups are centrally nilpotent of class at most 2.     

K-loops and Bruck loops on ℝ×ℝ

Using the function (x)=cosh x, K-loops on × are constructed. Since every K-loop is a Bruck loop, we have also examples for Bruck loops. Furthermore we investigate the group of the automorphisms _a,b of the K-loop which satisfy the equation a(bc)=(ab)_a,b(c).Dedicated to Professor Dr. Oswald Giering on the occasion of his 60 th birthday     

Symmetric designs with bruck subdesigns

IfP is a finite projective plane of ordern with a proper subplaneQ of orderm which is not a Baer subplane, then a theorem of Bruck [Trans. AMS 78(1955), 464–481] asserts thatn≧m ²+m. If the equalityn=m ²+m were to occur thenP would be of composite order andQ should be called a Bruck subplane. It can be shown that if a projective planeP contains a Bruck subplaneQ, then in factP contains a designQ′ which has the parameters of the lines in a three dimensional projective geometry of orderm. A well known scheme of Bruck suggests using such aQ′ to constructP. Bruck’s theorem readily extends to symmetric designs [Kantor, Trans. AMS 146 (1969), 1–28], hence the concept of a Bruck subdesign. This paper develops the analoque ofQ′ and shows (by example) that the analogous construction scheme can be used to find symmetric designs.     

K-loops from classical groups over ordered fields

K-loops will be constructed as transversals in classical groups over an ordered field. Manywell-known examples are subsumed in the present approach. Special attention is payed to the question which of these K-loops have fixed point free left inner mappings. Some new such examples are given. Isomorphisms between some of the K-loops are established as well.Dedicated to Professor Dr. Dr. h. c. Helmut Karzel on the occasion of his seventieth birthday     

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.     

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.     

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.     

Loops, their cores and symmetric spaces

This paper is devoted to the relations among affine symmetric spaces, smooth Bol and Moufang loops, smooth left distributive quasigroups and differentiable 3-nets. The results are used to prove the analyticity of smooth Moufang loops and left distributive quasigroups with involutive left translations as well as to show the Lie nature of transformation groups naturally related to some classes of smooth binary systems and 3-nets. In the last section we establish power series expansion for local loops with weak associativity conditions and apply the methods of the previous sections in order to describe geodesic loops having euclidean lines either as their geodesic lines or as geodesic lines of their core. The first author was partly supported by the Deutsche Forschungsgemeinschaft and by OTKA Grant no. T020545.     

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 Presentations of Bruck–Reilly Extensions

We first consider the class of monoids in which every left invertible element is also right invertible, and prove that if a monoid belonging to this class admits a finitely presented Bruck–Reilly extension then it is finitely generated. This allow us to obtain necessary and sufficient conditions for the Bruck–Reilly extensions of this class of monoids to be finitely presented. We then prove that thes 𝒟-classes of a Bruck–Reilly extension of a Clifford monoid are Bruck–Reilly extensions of groups. This yields another necessary and sufficient condition for these Bruck–Reilly extensions to be finitely generated and presented. Finally, we show that a Bruck–Reilly extension of a Clifford monoid is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid, and that this property cannot be generalized to Bruck–Reilly extensions of arbitrary inverse monoids.     

Construction of a class of loops via graphs

Starting from an involutorial difference loop (P, +) of order n we construct a new loop ${(L, \oplus)}$ in the same class and possessing 2n elements. The construction was induced by studying a correspondence connecting such loops with complete graphs with parallelism and regular involution sets (see “Basic notions and known results”). We discuss conditions on (P, + ) ensuring that the loop ${(L, \oplus)}$ is a K-loop and we give explicit examples of K-loops obtained with this method. Further generalizations of this technique are proposed as well.     

The André/Bruck and Bose representation of conics in Baer subplanes of PG(2,q 2)

The André/Bruck and Bose representation ([1], [5,6]) of PG(2,q ²) in PG(4,q) is a tool used by many authors in the proof of recent results. In this paper the André/Bruck and Bose representation of conics in Baer subplanes of PG(2,q ²) is determined. It is proved that a non-degenerate conic in a Baer subplane of PG(2,q ²) is a normal rational curve of order 2, 3, or 4 in the André/Bruck and Bose representation. Moreover the three possibilities (classes) are examined and we classify the conics in each class. Received 1 September 1999; revised 17 July 2000.