Local half Moufang quadrangles

In this article, we show that if every root of a finite generalized quadrangle containing a fixed point x is Moufang, then every dual root containing x in its interior is also Moufang. As a corollary, we obtain a new proof of the half Moufang theorem. This says that finite half Moufang quadrangles are Moufang.     

Polarities and planar collineations of Moufang planes

We show that conjugacy classes of Baer involutions and non-elliptic polarities, respectively, of proper (i.e., non-desarguesian) Moufang planes are interrelated. Restriction of the conjugating group to the stabilizer of a triangle or a quadrangle does not refine the classes. These results are applied to prove transitivity properties for the centralizers of these polarities. Along the way, a new proof is obtained for the fact that the automorphism group of a Moufang plane acts transitively on quadrangles.     

Moufang loops with commuting inner mappings

We investigate the relation between the structure of a Moufang loop and its inner mapping group. Moufang loops of odd order with commuting inner mappings have nilpotency class at most 2. The 6-divisible Moufang loops with commuting inner mappings have nilpotency class at most 2. There is a Moufang loop of order 2¹⁴ with commuting inner mappings and of nilpotency class 3.     

Moufang sets and jordan division algebras

We give a simple criterion which determines when a permutation group U and one additional permutation give rise to a Moufang set. We apply this criterion to show that every Jordan division algebra gives rise in a very natural way to a Moufang set, to provide sufficient conditions for a Moufang set to arise from a Jordan division algebra and to give a characterization of the projective Moufang sets over a commutative field of characteristic different from 2. The first author is a Postdoctoral Fellow of the Research Foundation – Flanders (Belgium) (FWO-Vlaanderen).     

Continuous Moufang Transformations

Continuous Moufang transformations are introduced and discussed. Commutation relations for infinitesimal Moufang transformations are established. The resulting Lie algebra has quite impressive structure equations, well known from the theory of alternative algebras.     

Moufang sets from groups of mixed type

We consider the existence of Moufang sets related to certain groups of mixed type. This way, we obtain new examples of Moufang sets and new constructions of known classes. The most interesting class of new examples is related to the Moufang quadrangles of type and to the Ree–Tits octagon over a nonperfect field, and the root groups of each member have nilpotency class three.     

On Centrally and Nuclearly Nilpotent Moufang Loops

Let Q be a finite Moufang loop with nucleus N(Q) and associator subloop 𝒜(Q). First we prove if the factor loop over the nucleus Q/N has nontrivial center, then the center of Q is nontrivial too. By using this result we prove that the centrally nilpotence of Q/N(Q) implies the centrally nilpotence of 𝒜(Q), and we show that, for the centrally nilpotence of a finite Moufang loop, the centrally nilpotence of Q/N(Q) and Q/𝒜(Q) is a necessary and sufficient condition. Finally, as a corollary we give a necessary and sufficient condition for the equivalence of centrally and nuclearly nilpotence of finite Moufang loops, namely, the centrally nilpotence of Q/𝒜(Q).     

Sylow's theorem for Moufang loops

For finite Moufang loops, we prove an analog of the first Sylow theorem giving a criterion for the existence of a p-Sylow subloop. We also find the maximal order of p-subloops in the Moufang loops that do not possess p-Sylow subloops.     

Universal envelopes of Malcev Algebras: Pointed Moufang bialgebras

Under study are the pointed unital coassociative cocommutative Moufang H-bialgebras. We prove an analog of the Cartier-Kostant-Milnor-Moore theorem for weakly associative Moufang H-bialgebras. If the primitive elements commute with group-like elements then these Moufang H-bialgebras are isomorphic to the tensor product of a universal enveloping algebra of a Malcev algebra and a loop algebra constructed by a Moufang loop.     

On 4-transitivity in the Moufang plane

The purpose of this paper is to give a short proof of 4-transitivity in Moufang planes. This proof originated in the observation of the two first name authors that the standard Moufang identities, together with the identity (1) x^–1(y(xz)) = (x^–1(yx)z, which is asserted in [2, p. 103] to hold in Cayley-Dickson division algebras, can be applied to give a particularly simple algebraic proof of the fact that the collineation group of a Moufang plane is transitive on four-points. Unfortunately, as pointed out by H. Karzel and demonstrated here in Proposition 1, (1) does not hold in Cayley-Dickson algebras. Nevertheless, the algebraic proof of transitivity remains valid after slight modifications and is given here as Theorem 1.The authors wish to thank Professor Karzel for pointing out the error in [2] and for his suggestions in preparing the final version of this paper.Dedicated to Professor H. Karzel on the occasion of his 60th birthday.     

Half-isomorphisms of Finite Automorphic Moufang Loops

We show that each half-automorphism of a finite automorphic Moufang loop is trivial. In general, this is not true for finite left automorphic Moufang loops and for finite automorphic loops.     

On the Bruck-Slaby theorem for commutative Moufang loops

With the help of the relationship between commutative Moufang loops and alternative commutative algebras, we prove, rather easily, the following weakened version of the Bruck-Slaby theorem: a finitely generated commutative Moufang loop is centrally nilpotent. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 275–281, August, 1999.     

Abelian-by-Cyclic Moufang Loops

We use groups with triality to construct a series of nonassociative Moufang loops. Certain members of this series contain an abelian normal subloop with the corresponding quotient being a cyclic group. In particular, we give a new series of examples of finite abelian-by-cyclic Moufang loops. The previously known [10 Rajah , A. ( 2001 ). Moufang loops of odd order pq ³ . J. Algebra 235 ( 1 ): 66 – 93 .[Crossref], [Web of Science ®] , [Google Scholar]] loops of this type of odd order 3q ³, with prime q ≡ 1 (mod 3), are particular cases of our series. Some of the examples are shown to be embeddable into a Cayley algebra.     

Every diassociative A-loop is Moufang

An A-loop is a loop in which every inner mapping is an automorphism. A problem which had been open since 1956 is settled by showing that every diassociative A-loop is Moufang.

     

Minimally Nonassociative Commutative Moufang Loops

A commutative Moufang loop which is not associative has the property that all proper subloops are associative if and only if the loop can be generated by three elements.     

A weak Moufang condition suffices

If Γ is a 2-Moufang generalized n-gon for n≤6, then Γ is Moufang.     

Commutative Moufang loops and Bessel functions

LetF be a field. For eachk>1, letG be a finite group containing{x ₁,...,x _k }!×{y ₁,...,y _k}!. Then in the group algebraFG, $$co\dim _F \sum\limits_{j = 1}^{k - 1} {(1 + (x_j x_{j + 1} ))(1 + (y_j y_{j + 1} ))FG = \frac{{|G|}}{{2\pi i}}\oint\limits_{|z| = 1} {\frac{{dz}}{{J_0 (2\sqrt z )z^{k + 1} }}.} } $$ Connections with the theory of commutative Moufang loops are discussed, including a conjectured answer to Manin's problem of specifying the 3-rank of a finitely generated free commutative Moufang loop.     

Every Moufang loop of odd order with nontrivial commutant has nontrivial center

We get a partial result for Phillips’ problem: does there exist a Moufang loop of odd order with trivial nucleus? First we show that a Moufang loop Q of odd order with nontrivial commutant has nontrivial nucleus, then, by using this result, we prove that the existence of a nontrivial commutant implies the existence of a nontrivial center in Q. Introducing the notion of commutantly nilpotence, we get that the commutantly nilpotence is equivalent to the centrally nilpotence for the Moufang loops of odd order.