Abstract. A loop 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, and the order in which the variables appear on both sides is the same, viz. ((xy)x)z = x(y(xz)). Loop varieties defined by one identity of Bol-Moufang type include groups, Bol loops, Moufang loops and C-loops. We show that there are exactly 14 such varieties, and determine all inclusions between them, providing all necessary counterexamples, too. This extends and completes the programme of Fenyves [5].

Let Q be a conjugacy closed loop, and N (Q) its nucleus. Then Z(N (Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is power-associative and |Q| is finite and relatively prime to 6, then Q is a group. If Q is a finite non-associative extra loop, then 16 | |Q|. * Author supported by NSF Grant DMS-0097881 2 Proof. RCC and LCC assert that L y L x = L x L f (x,y) and R y R x = R x R g(x,y) .Thus, in a CC-loop, the left multiplications are closed under conjugation and the right multiplications are closed under conjugation; hence the name "conjugacy-closed".These loops have a number of interesting properties, surveyed in Sections 2 and 3; for example, by [10], the left and right inner mappings are automorphisms. These properties allow a rather detailed structural analysis to be made; in particular, all CC-loops of orders p 2 and 2p (for primes p) are known (see [13]). This paper yields additional structural information about CC-loopsespecially for the ones which are power-associative (that is, each x is a group) or diassociative (that is, each x, y is a group).It is shown in [11] that the CC-loops which are diassociative (equivalently, Moufang) are the extra loops studied by Fenyves [8,9]. By [9], if Q is an extra loop, then Q/N(Q) is a boolean group (where N(Q) is the nucleus). It is immediate that a finite extra loop of odd order is a group. We show here (Corollary 7.7) that a finite power-associative CC-loop of order relatively prime to 6 is a group. The "6" cannot be improved, since there are non-associative power-associative CC-loops of order 16 (e.g., the Cayley loop) and of order 27 (see Section 9) (we do not know if there are ones of order divisible by 6 but not by 4 or 9). Also, one cannot drop the "power-associative", since by [10], there are non-power-associative CC-loops of order p 2 for every odd prime p.More generally, we show that every power-associative CC-loop satisfies a weakening of diassociativity -namely, x, y is a group whenever x is a cube and y is a square. Then, if |Q| is relatively prime to 6, every element must be a sixth power by the Lagrange property, so that Q is diassociative, and hence an extra loop of odd order, and hence a group. Of course, we must verify that the Lagrange property really holds for CC-loops, since it can fail for loops in general. This is easy to do (see Corollary 3.2) using the result of Basarab [2]. He showed that for any CC-loop, Q/N(Q) is an abelian group (this answers a question from [10]); we include a proof of this here (see Theorem 3.1), since it is fairly short using the notion of autotopy (see Belousov [3] II §3 or Bruck [5] VII §2), together with some facts about the autotopies of CC-loops proved by Goodaire and Robinson (see [10] and Section 2).We also establish two theorems about general CC-loops. First, whenever S ⊆ Q and S associates in the sense that x · yz = xy · z holds for all x, y, z ∈ S, we prove that S also associates, and hence is a group (see Cor...

We solve a problem of Belousov which has been open since 1967: to characterize the loop isotopes of F-quasigroups. We show that every F-quasigroup has a Moufang loop isotope which is a central product of its nucleus and Moufang center. We then use the loop to reveal the structure of the associated F-quasigroup.

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. ((xy)x)z = x(y(xz)). 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 computergenerated.

Abstract. Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops.We study uniquely 2-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman's work on uniquely 2-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop Q of odd order is solvable, contains an element of order p for every prime p dividing |Q|, and |S| divides |Q| for every subloop S of Q.There are no finite simple nonassociative commutative automorphic loops, and there are no finite simple nonassociative automorphic loops of order less than 2500. We show that if Q is a finite simple nonassociative automorphic loop then the socle of the multiplication group of Q is not regular. The existence of a finite simple nonassociative automorphic loop remains open.Let p be an odd prime. Automorphic loops of order p or p 2 are groups, but there exist nonassociative automorphic loops of order p 3 , some with trivial nucleus (center) and of exponent p. We construct nonassociative "dihedral" automorphic loops of order 2n for every n > 2, and show that there are precisely p − 2 nonassociative automorphic loops of order 2p, all of them dihedral.