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We prove that a normal subloop X of a Moufang loop Q induces an abelian congruence of Q if and only if u (x y) = (u y) x for all x , y ∈ X and u ∈ Q . This characterization is then used to show that classically solvable finite 3-divisible Moufang loops are congruence solvable. [ABSTRACT FROM AUTHOR]

Let Q be a loop. The mappings x ↦ a x , x ↦ x a and x ↦ a / x are denoted by L a , R a and D a , respectively. The loop is said to be middle Bruck if for all a , b ∈ Q there exists c ∈ Q such that D a D b D a = D c . The right inverse of Q is the loop with operation x / (y \ 1) . It is proved that Q is middle Bruck if and only if the right inverse of Q is left Bruck (i.e., a left Bol loop in which (x y) - 1 = x - 1 y - 1 ). Middle Bruck loops are characterized in group theoretic language as transversals T to H ≤ G such that ⟨ T ⟩ = G , T G = T and t 2 = 1 for each t ∈ T . Other results include the fact that if Q is a finite loop, then the total multiplication group ⟨ L a , R a , D a ; a ∈ Q ⟩ is nilpotent if and only if Q is a centrally nilpotent 2-loop, and the fact that total multiplication groups of paratopic loops are isomorphic. [ABSTRACT FROM AUTHOR]

A loop X is said to satisfy Moufang's theorem if for every x , y , z ∈ X such that x (y z) = (x y) z the subloop generated by x, y, z is a group. We prove that the variety V of Steiner loops satisfying the identity (x z) (((x y) z) (y z)) = ((x z) ((x y) z)) (y z) is not contained in the variety of Moufang loops, yet every loop in V satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah. [ABSTRACT FROM AUTHOR]

Let Q be a quasigroup. For α,β∈SQ let Qα,β be the principal isotope x∗y=α(x)β(y). Put a(Q)=|{(x,y,z)∈Q3;x(yz))=(xy)z}| and assume that |Q|=n. Then ∑α,βa(Qα,β)/(n!)2=n2(1+(n-1)-1), and for every α∈SQ there is ∑βa(Qα,β)/n!=n(n-1)-1∑x(fx2-2fx+n)≥n2, where fx=|{y∈Q;y=α(y)x}|. If G is a group and α is an orthomorphism, then a(Gα,β)=n2 for every β∈SQ. A detailed case study of a(Gα,β) is made for the situation when G=Z2d, and both α and β are “natural” near-orthomorphisms. Asymptotically, a(Gα,β)>3n if G is an abelian group of order n. Computational results: a(7)=17 and a(8)≤21, where a(n)=min{a(Q);|Q|=n}. There are also determined minimum values for a(Gα,β), G a group of order ≤8. [ABSTRACT FROM AUTHOR]

We present equivalent definitions of code loops in any characteristic p≠0. The most natural definition is via combinatorial polarization, but we also show how to realize code loops by linear codes and as a class of symplectic conjugacy closed loops. For p odd, it is possible to define code loops via characteristic trilinear forms. Related concepts are discussed. [ABSTRACT FROM AUTHOR]

We consider the isomorphism problem for quasigroups isotopic to a group G and show that the isomorphism classes correspond to orbits of an action by the holomorph of G upon pairs (α, β), where α and β permute G and α(1) = 1. The results are applied to central quasigroups and to quasigroups of order 4. We also characterize equationally those isotopes of a group G in which α ∈ Aut G. [ABSTRACT FROM AUTHOR]

We examine two particular constructions that derive from a 2-group G= G(⋅) another 2-group G(*) for the case when G(⋅) is one of $D_{2^{n}}$ , $SD_{2^{n}}$ , $Q_{2^{n}}$ . The constructions (cyclic and dihedral) have the property that x* y= x⋅ y for exactly 3/4 of all pairs ( x, y)∈ G× G. [ABSTRACT FROM AUTHOR]

Several new tools are developed dealing with multiplication groups of loops. One of them is based upon the theorem that the inner mapping group of a loop never acts faithfully on any of its orbits, unless it is trivial. The applications include an analysis of loops whose inner mapping group is nonabelian of order pq. Some important earlier results are proved anew in a shorter way. [ABSTRACT FROM AUTHOR]

An equivalence ∼ upon a loop is said to be multiplicative if it satisfies x ∼ y , u ∼ v ⇒ x u ∼ y v . Let X be a set with elements x ≠ y and let ∼ be the least multiplicative equivalence upon a free loop F(X) for which x ∼ y . If a , b ∈ F (X) are such that a ≠ b and a ∼ b , then neither a \ c ∼ b \ c nor c / a ∼ c / b is true, for every c ∈ F (X) . [ABSTRACT FROM AUTHOR]