Lindner C.C. Quasigroups constructed from cycle systems

An m-cycle system of order n is a partition P of the edges of Kn, each element of which induces a cycle of length m. An m-cycle system P is said to be i-perfect if each pair of vertices is distance i apart in exactly one cycle in P. Given an m-cycle system P, one can define an idempotent binary operation using the standard construction; namely, for different i, j, the operation is defined by i·j=k if and only if the cycle (...,i,j,k,...) is in P. If P is 2-perfect then this binary operation defines a quasigroup that satisfies various identities. One can then consider all the finite quasigroups that satisfy these identities and see if they arise from an $m$-cycle system in this manner; if so, then the set of 2-perfect m-cycle systems is said to be equationally defined. This paper gives a survey of results on this topic. In particular, it is shown that the set of 2-perfect m-cycle systems is equationally defined if and only if m=3,5 or 7. The paper also gives a survey of related results for m-perfect (2m+1)-cycle systems, and for 2-perfect directed m-cycle systems. ( Chris Rodger, MR 2004e:05033 ) Back

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Nagy G.P., Vojtĕchovský P. Octonions, simple Moufang loops and triality

This is a very interesting and informative paper in which the authors expound upon the connections between a number of different concepts in loop theory to produce a new proof, using geometric loop theory, of the well known result that the nonassociative finite simple Moufang loops are precisely the loops constructed by L. J. Paige [Proc. Amer. Math. Soc. 7 (1956), 471--482; MR 18, 110f] from Zorn vector matrix algebras. Chapter 2 contains discussions of the connections between loops and 3-nets, between collineations of nets and autotopisms of loops, and between Moufang loops, Moufang 3-nets, and the set of Bol reflections of a 3-net. Chapter 3 deals with composition algebras, the Cayley-Dickson process, and the split octonion algebra, O, over a field F. In Chapter 4, the Paige loops M*(F) are constructed and their multiplication groups are found. Chapter 5 considers connections between Moufang 3-nets and groups with triality - groups with a set S of automorphisms isomorphic to the symmetric group S3, in which a certain "triality identity" holds. Starting with a Moufang 3-net, one can associate a group with triality, and conversely. Homomorphisms between groups with triality which "preserve" the respective triality maps are known as S-homomorphisms, and the kernel of such a homomorphism is an S-invariant normal subgroup. Groups with triality with no such nontrivial S-invariant subgroups are S-simple. Chapter 6 completes the classification of finite simple Moufang loops, by first showing that if L is a simple Moufang loop, then the associated 3-net and hence the group with triality, (G,S), that it determines is S-simple, and by then classifying the S-simple groups with triality. Chapter 7 discusses the automorphism groups of the Paige loops in the case that F is a perfect field, and Chapter 8 presents some results on the generators of the Paige loops and poses some conjectures and open problems. ( Orin Chein, MR 2004f:20118 ) Back

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Phillips J. D. See Otter digging for algebraic pearls

Otter (devised by W. McCune), by carefully and critically analysing its features via a quite readable survey, which, although not a manual (in the author's own words), may prove to be quite useful to loop theorists. The basic idea is to exploit Otter's equational reasoning in analysing old and open problems in loop theory. The proofs obtained with Otter's assistance need translation into a "human" code, due to their length and opaqueness (this applies even to easy ones). This notwithstanding, the great heuristic value of this kind of approach is undeniable, as the author duly documents, by showing many interesting by-products of the translation process. The paper is organized as follows: after a presentation of Otter, there follow an input file together with an enhanced one, the associated proof accompanied by translation, and then a critical analysis thereof. New results and perspectives close the survey. ( Elena Zizioli, MR 2004g:20090 ) Back

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Smith J. D. H. Quasigroup permutation representations

The article surveys the theory of permutation representations of finite quasigroups. Quasigroups are given as sets (Q, . , / , \ ) endowed with three binary operations: multiplication . , right division / and left division \ . The concept of homogeneous space for finite quasigroups is analogous to that for groups. For a given subquasigroup P of a finite quasigroup Q the elements of the corresponding homogeneous space P\Q are the orbits on Q of the group of permutations generated by the left multiplications by elements of P. Each element of Q yields a Markov chain action on the homogeneous space P\Q as a set of states. Moreover, the full structure is an instance of an iterated function system (IFS) in the sense of fractal geometry. For associative quasigroups this concept of homogeneous space specializes to the well-known permutation representation of a group, where the transition matrices of the Markov chains become permutation matrices. For a finite set Q, the Q-IFS are realised as coalgebras for the Q-th power of the endofunctor B sending a set to the underlying set of the free barycentric algebra it generates. Thus each homogeneous space over a finite quasigroup Q yields a BQ-algebra and the class of all permutation representations ("Q-sets") of a given fixed quasigroup Q forms a covariety of coalgebras. Burnside's lemma extends to quasigroup permutation representations and has the following form: Let $X$ be a finite Q-set over a finite, non-empty quasigroup Q. Then the trace of the Markov matrix of X is equal to the number of orbits of X. The theory of permutation representations of finite quasigroups offers a new approach to the study of Lagrangian properties of loops. The right Lagrange property is inherited by subloops and homomorphic images. It is an open problem whether the product of two finite loops Q1 and Q2 satisfies the right Lagrangian property if Q1 and Q2 do. ( Huberta Lausch, MR 2004i:20127 ) Back

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