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 Syl...

Moufang’s theorem states that if Q is a Moufang loop with elements x, y and z such that x ·yz = xy · z, then these three elements generate a subgroup of Q. The paper contains a new proof of this theorem that is shorter and more transparent than the standardly used proof of Bruck. A loop is a binary structure with a unit such that the equations ax = b and ya = b have unique solutions x = a\b and...

The probability that m randomly chosen elements of a finite power associative loop C have prescribed orders and generate C is calculated in terms of certain constants Γ related to the action of Aut(C) on the subloop lattice of C. As an illustration, all meaningful probabilities of random generation by elements of given orders are found for the smallest nonassociative simple Moufang loop. 1. Ran...

Let Q be a Buchsteiner loop. We describe the associator calculus in three variables, and show that |Q| ≥ 32 if Q is not conjugacy closed. We also show that |Q| ≥ 64 if there exists x ∈ Q such that x is not in the nucleus of Q. Furthermore, we describe a general construction that yields all proper Buchsteiner loops of order 32. Finally, we produce a Buchsteiner loop of order 128 that is nilpoten...

We show that in a weak commutative inverse property loop, such as a Bruck loop, if α is a right [left] pseudoautomorphism with companion c, then c [c] must lie in the left nucleus. In particular, for any such loop with trivial left nucleus, every right pseudoautomorphism is an automorphism and if the squaring map is a permutation, then every left pseudoautomorphism is an automorphism as well. W...

A loop Q is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. Loops in which the left and right nuclei coincide and are of index 2 are necesarilly LCC, and they are constructed in the paper explicitly. LCC loops Q with the right nucleus G of index 2 offer a larger diversity. A sample of results: if Z(G) = 1, then Q is al...

We study conjugacy closed loops by means of their multiplication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication groups, respectively. Put M = {a ∈ Q; La ∈ R}. We prove that the cosets of A agree with orbits of [L,R], that Q/M ∼= (InnQ)/L1 and that one can define an abelian group on Q/N × Mlt1. We also explain why ...

A groupoid is alternative if it satisfies the alternative laws x(xy) = (xx)y and x(yy) = (xy)y. These laws induce four partial maps on N+ × N+ (r, s) 7→ (2r, s− r), (r − s, 2s), (r/2, s + r/2), (r + s/2, s/2), that taken together form a dynamical system. We describe the orbits of this dynamical system, which allows us to show that nth powers in a free alternative groupoid on one generator are w...

We start by describing all the varieties of loops Q that can be defined by autotopisms αx, x ∈ Q, where αx is a composition of two triples, each of which becomes an autotopism when the element x belongs to one of the nuclei. In this way we obtain a unifying approach to Bol, Moufang, extra, Buchsteiner and conjugacy closed loops. We reprove some classical facts in a new way and show how Buchstei...