Moufang loop

About: Moufang loop is a research topic. Over the lifetime, 217 publications have been published within this topic receiving 1802 citations.

An Introduction to Quasigroups and Their Representations

Abstract: QUASIGROUPS AND LOOPS Latin squares Equational quasigroups Conjugates Semisymmetry and homotopy Loops and piques Steiner triple systems I Moufang loops and octonions Triality Normal forms Exercises Notes MULTIPLICATION GROUPS Combinatorial multiplication groups Surjections The diagonal action Inner multiplication groups of piques Loop transversals and right quasigroups Loop transversal codes Universal multiplication groups Universal stabilizers Exercises Notes CENTRAL QUASIGROUPS Quasigroup congruences Centrality Nilpotence Central isotopy Central piques Central quasigroups Quasigroups of prime order Stability congruences No-go theorems Exercises Notes HOMOGENEOUS SPACES Quasigroup homogeneous spaces Approximate symmetry Macroscopic symmetry Regularity Lagrangean properties Exercises Notes PERMUTATION REPRESENTATIONS The category IFSQ Actions as coalgebras Irreducibility The covariety of Q-sets The Burnside algebra An example Idempotents Burnside's lemma Exercises Problems Notes CHARACTER TABLES Conjugacy classes Class functions The centralizer ring Convolution of class functions Bose-Mesner and Hecke algebras Quasigroup character tables Orthogonality relations Rank two quasigroups Entropy Exercises Problems Notes COMBINATORIAL CHARACTER THEORY Congruence lattices Quotients Fusion Induction Linear characters Exercises Problems Notes SCHEMES AND SUPERSCHEMES Sharp transitivity More no-go theorems Superschemes Superalgebras Tensor squares Relation algebras The reconstruction theorem Exercises Problems Notes PERMUTATION CHARACTERS Enveloping algebras Structure of enveloping algebras The canonical representation Commutative actions Faithful homogeneous spaces Characters of homogeneous spaces General permutation characters The Ising model Exercises Problems Notes MODULES Abelian groups and slice categories Quasigroup modules The fundamental theorem Differential calculus Representations in varieties Group representations Exercises Problems Notes APPLICATIONS OF MODULE THEORY Nonassociative powers Exponents Steiner triple systems II The Burnside problem A free commutative Moufang loop Extensions and cohomology Exercises Problems Notes ANALYTICAL CHARACTER THEORY Functions on finite quasigroups Periodic functions on groups Analytical character theory Almost periodic functions Twisted translation operators Proof of the existence theorem Exercises Problems Notes APPENDIX A: CATEGORICAL CONCEPTS Graphs and categories Natural transformations and functors Limits and colimits APPENDIX B: UNIVERSAL ALGEBRA Combinatorial universal algebra Categorical universal algebra APPENDIX C: COALGEBRAS Coalgebras and covarieties Set functors REFERENCES INDEX

Moufang Loops of Small Order

Abstract: The main result of this paper is the determination of all nonassociative Moufang loops of orders *31. Combinatorial type methods are used to consider a number of cases which lead to the discovery of 13 loops of the type in question and prove that there can be no others. All of the loops found are isomorphic to all of their loop isotopes, are solvable, and satisfy both Lagrange's theorem and Sylow's main theorem. In addition to finding the loops referred to above, we prove that Moufang loops of orders p, p , p or pq (for p and q prime) must be groups. Finally, a method is found for constructing nonassociative Moufang loops as extensions of nonabelian groups by the cyclic group of order 2. L Introduction. In studying algebraic objects, it is frequently useful to have many examples at one's fingertips. In the case of Moufang loops that are not groups, the scarcity of manageable examples is one of the difficulties that we have encountered. It is the purpose of this paper to begin to remedy this situation by finding all Moufang loops of order < 31.0) There are 13 such loops-one of order 12, five of order 16, one of order 20, five of order 24, and one of order 28. The order structures, nuclei and subloops of these loops are given (Tables 3, 4 and 5). All of the loops are G-loops (i.e. they are isomorphic to all of their loop isotopes) and they are solvable. Lagrange's theorem and Sylow's main theorem hold in all of them. In terms of the M^-laws of Pflugfelder [10], some of the loops are M}-loops, some are M7-loops, and some are strictly Moufang. In the course of studying these loops, we find a general method of constructing nonassociative Moufang loops as extensions of groups (see Theorem 1). We also prove that, for p and q being primes, Moufang loops of order pq or of order p" for n < 3 are groups. Presented to the Society, July 15, 1971; received by the editors November 15, 1971. AUS (MOS) subject classifications (1970). Primary 20N05.

Simple Moufang loops

Abstract: If H is a Moufang loop, and x ∈ H, there are defined permutations of H, L(x):y ↦ xy and R(x): y ↦ yx. The group Gr (H), generated by these permutations for all choices of x, is called the multiplication group of H. It has a close connexion with the structure of H, as shown, for instance, in the papers of Albert(1). The purpose of this paper is to investigate the correspondence between groups and loops, so that group theoretic results may be applied to determine the structure of Moufang loops.

The classification of finite simple Moufang loops

Abstract: The purpose of this paper is to classify the finite simple Moufang loops. A Moufang loop M is a loop which satisfies the identitynote that the equivalent identities ((xy)z)y = x(y(zy)), x(y(xz)) = ((xy)x)z also hold, by [2], p. 115. The Moufang loop M is simple if it has no non-trivial proper homomorphic images, or equivalently, if it has no non-trivial proper normal subloops. For basic definitions and properties of Moufang loops, see [2] – in particular, the Jordan–Holder theorem holds for finite Moufang loops ([2], p. 67). Of course if the finite simple loop M is associative, then M is a simple group, and hence is determined by the classification of finite simple groups. In [9], Paige defines, for each finite field GF(q), a finite simple Moufang loop M(q) which is not associative – M(q) is essentially the set of units in the eight-dimensional split Cayley algebra over GF(q), modulo the centre (we shall describe M(q) in much more detail in §2).

A class of simple Moufang loops

Abstract: As we shall see in the course of our proof, the present theorem is a nonassociative analogue of the well known results on the special projective group PSL(n, K) (see [4, p. 44]). In ?5, we shall prove that the Cayley-Dickson numbers of norm 1 over the real field R* (modulo their center) are simple and indicate how this is the best possible result. Our results will yield finite, not-associative, simple Moufang loops whose possible orders are (21n-22n) and 2-1(p7n_ p3n) if p is an odd prime. Thus we obtain a simple, not-associative, Moufang loop of order 120. Although we have tried to make this paper reasonably self contained, some of the results by Bruck (2) on Moufang loops will be used without reference.