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In groups, an abelian normal subgroup induces an abelian congruence. We construct a class of centrally nilpotent Moufang loops containing an abelian normal subloop that does not induce an abelian congruence. On the other hand, we prove that in 6-divisible Moufang loops, every abelian normal subloop induces an abelian congruence. In loops, congruence solvability adopted from the universal-algebraic commutator theory of congruence modular varieties is strictly stronger than classical solvability adopted from group theory. It is an open problem whether the two notions of solvability coincide in Moufang loops. We prove that they coincide in 6-divisible Moufang loops and in Moufang loops of odd order. In fact, we show that every Moufang loop of odd order is congruence solvable, thus strengthening Glauberman's Odd Order Theorem for Moufang loops. [ABSTRACT FROM AUTHOR]

Let p > q be odd primes. We classify Bol loops and Bruck loops of order pq up to isotopism. When q does not divide p 2 − 1 , the only Bol loop (and hence the only Bruck loop) of order pq is the cyclic group of order pq . When q divides p 2 − 1 , there are precisely ⌊ ( p − 1 + 4 q ) ( 2 q ) − 1 ⌋ Bol loops of order pq up to isotopism, including a unique nonassociative Bruck loop of order pq . [ABSTRACT FROM AUTHOR]

We present an algorithm for calculating a minimal generating set of a finite algebra. Despite the fact that the problem is in NP, a single call to a SAT solver is impractical since the encoding is cubic. Instead, the proposed algorithm solves a series of smaller subproblems. The individual subproblems are formulated as integer linear programs (ILP) that are solved by an off-the-shelf solver. Our implementation shows that the proposed algorithm is highly efficient and is able to compute minimal generators for algebras of orders approximately 2000. In our experiments we focus on Moufang loops, a variety of loops with properties close to groups. For Moufang loops of prime power order, we are able to calculate a minimal generating set by another method, using theoretical results on the Frattini subloop and algorithms for permutation groups, of which some are reported here for the first time. This second method does not cover all cases, but in the covered cases it serves as a check of correctness of the ILP-based algorithm. [ABSTRACT FROM AUTHOR]

Code loops are certain Moufang 2-loops constructed from doubly even binary codes that play an important role in the construction of local subgroups of sporadic groups. More precisely, code loops are central extensions of the group of order 2 by an elementary abelian 2-group V in the variety of loops such that their squaring map, commutator map and associator map are related by combinatorial polarization and the associator map is a trilinear alternating form. Using existing classifications of trilinear alternating forms over the field of 2 elements, we enumerate code loops of dimension d = dim ⁡ ( V ) ≤ 8 (equivalently, of order 2 d + 1 ≤ 512 ) up to isomorphism. There are 767 code loops of order 128, and 80826 of order 256, and 937791557 of order 512. [ABSTRACT FROM AUTHOR]

Abstract: Using the Freese–McKenzie commutator theory for congruence modular varieties as the starting point, we develop commutator theory for the variety of loops. The fundamental theorem of congruence commutators for loops relates generators of the congruence commutator to generators of the total inner mapping group. We specialize the fundamental theorem into several varieties of loops, and also discuss the commutator of two normal subloops. Consequently, we argue that some standard definitions of loop theory, such as elementwise commutators and associators, should be revised and linked more closely to inner mappings. Using the new definitions, we prove several natural properties of loops that could not be so elegantly stated with the standard definitions of loop theory. For instance, we show that the subloop generated by the new associators defined here is automatically normal. We conclude with a preliminary discussion of abelianess and solvability in loops. [Copyright &y& Elsevier]

Abstract: Suppose that all groups of order n are defined on the same set G of cardinality n, and let the distance of two groups of order n be the number of pairs where the two group operations differ. Given a group of order n, we find all groups of order n, up to isomorphism, that are closest to . [Copyright &y& Elsevier]

Abstract: The isomorphism problem for centrally nilpotent loops can be tackled by methods of cohomology. We develop tools based on cohomology and linear algebra that either lend themselves to direct count of the isomorphism classes (notably in the case of nilpotent loops of order 2q, q a prime), or lead to efficient classification computer programs. This allows us to enumerate all nilpotent loops of order less than 24. [Copyright &y& Elsevier]

Abstract: In 2004, Csörgő constructed a loop of nilpotency class three with abelian group of inner mappings. As of now, no other examples are known. We construct many such loops from groups of nilpotency class two by replacing the product with in certain positions, where is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form. [Copyright &y& Elsevier]

Abstract: We classify Moufang loops of order 64 and 81 up to isomorphism, using a linear algebraic approach to central loop extensions. In addition to the 267 groups of order 64, there are 4262 nonassociative Moufang loops of order 64. In addition to the 15 groups of order 81, there are 5 nonassociative Moufang loops of order 81, 2 of which are commutative. [Copyright &y& Elsevier]

Abstract: We show how to obtain all nonassociative Moufang loops of order less than 64 and 4262 nonassociative Moufang loops of order 64 in a unified way. We conjecture that there are no other nonassociative Moufang loops of order 64. The main idea of the computer search is to modify precisely one quarter of the multiplication table in a certain way, previously applied to small 2-groups. [Copyright &y& Elsevier]

Abstract: A quasigroup identity is of Bol–Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, the order in which the variables appear on both sides is the same, and the only binary operation used is the multiplication, viz. . Many well-known varieties of quasigroups are of Bol–Moufang type. We show that there are exactly 26 such varieties, determine all inclusions between them, and provide all necessary counterexamples. We also determine which of these varieties consist of loops or one-sided loops, and fully describe the varieties of commutative quasigroups of Bol–Moufang type. Some of the proofs are computer-generated. [Copyright &y& Elsevier]

Abstract: A groupoid identity is said to be linear of length if the same variables appear on both sides of the identity exactly once. We classify and count all varieties of groupoids defined by a single linear identity. For , there are 14 nontrivial varieties and they are in the most general position with respect to inclusion. Hentzel et al. [Hentzel, I.R., Jacobs, D.P., Muddana, S.V., 1993. Experimenting with the identity . J. Symbolic Comput. 16, 289–293] showed that the linear identity implies commutativity and associativity in all products of at least five factors. We complete their project by showing that no other linear identity of any length behaves this way, and by showing how the identity affects products of fewer than five factors; we include distinguishing examples produced by the finite model builder Mace4. The corresponding combinatorial results for labelled binary trees are given. We associate a certain wreath product with any linear identity. Questions about linear groupoids can therefore be transferred to groups and attacked by group-theoretical computational tools, e.g., GAP. Systematic notation and diagrams for linear identities are devised. A short equational basis for Boolean algebras involving the identity is presented, together with a proof produced by the automated theorem prover OTTER. [Copyright &y& Elsevier]

Right Bol loops are loops satisfying the identity ( ( z x ) y ) x = z ( ( x y ) x ) , and right Bruck loops are right Bol loops satisfying the identity ( x y ) − 1 = x − 1 y − 1 . Let p and q be odd primes such that p > q . Advancing the research program of Niederreiter and Robinson from 1981, we classify right Bol loops of order pq . When q does not divide p 2 − 1 , the only right Bol loop of order pq is the cyclic group of order pq . When q divides p 2 − 1 , there are precisely ( p − q + 4 ) / 2 right Bol loops of order pq up to isomorphism, including a unique nonassociative right Bruck loop B p , q of order pq . Let Q be a nonassociative right Bol loop of order pq . We prove that the right nucleus of Q is trivial, the left nucleus of Q is normal and is equal to the unique subloop of order p in Q , and the right multiplication group of Q has order p 2 q or p 3 q . When Q = B p , q , the right multiplication group of Q is isomorphic to the semidirect product of Z p × Z p with Z q . Finally, we offer computational results as to the number of right Bol loops of order pq up to isotopy. [ABSTRACT FROM AUTHOR]

We prove that if the squaring map in the factor loop of a Moufang loop Q over its nucleus is surjective, then every half-isomorphism of Q onto a Moufang loop is either an isomorphism or an anti-isomorphism. This generalizes all earlier results in this vein. [ABSTRACT FROM AUTHOR]

Abstract: A loop is automorphic if its inner mappings are automorphisms. Using so-called associated operations, we show that every commutative automorphic loop of odd prime power order is centrally nilpotent. Starting with suitable elements of an anisotropic plane in the vector space of matrices over the field of prime order p, we construct a family of automorphic loops of order with trivial center. [Copyright &y& Elsevier]

Wolfgang Rump showed that there is a one-to-one correspondence between nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation and binary algebras in which all left translations L x are bijections, the squaring map is a bijection, and the identity (x y) (x z) = (y x) (y z) holds. We call these algebras rumples in analogy with quandles, another class of binary algebras giving solutions of the Yang-Baxter equation. We focus on latin rumples, that is, on rumples in which all right translations are bijections as well. We prove that an affine latin rumple of order n exists if and only if n = p 1 p 1 k 1 ⋯ p m p m k m for some distinct primes p i and positive integers k i. A large class of affine solutions is obtained from nonsingular near-circulant matrices A , B satisfying [ A , B ] = A 2. We characterize affine latin rumples as those latin rumples for which the displacement group generated by L x L y − 1 is abelian and normal in the group generated by all translations. We develop the extension theory of rumples sufficiently to obtain examples of latin rumples that are not affine, not even isotopic to a group. Finally, we investigate latin rumples in which the dual identity (z x) (y x) = (z y) (x y) holds as well, and we show, among other results, that the generators L x L y − 1 of their displacement group have order dividing four. [ABSTRACT FROM AUTHOR]