Half Moufang implies Moufang for finite generalized quadrangles

Summary A finite generalized quadrangle has two types of panels. If each panel of one type is Moufang, then every panel is Moufang. Hence by a theorem of Fong and Seitz [1] the quadrangle is classical or dual classical.Oblatum 1-XI-1989 & 7-XI-1990     

On collineations and dualities of finite generalized polygons

In this paper we generalize a result of Benson to all finite generalized polygons. In particular, given a collineation θ of a finite generalized polygon S, we obtain a relation between the parameters of S and, for various natural numbers i, the number of points x which are mapped to a point at distance i from x by θ. As a special case we consider generalized 2n-gons of order (1, t) and determine, in the generic case, the exact number of absolute points of a given duality of the underlying generalized n-gon of order t.     

Moufang polygons and irreducible spherical BN-pairs of rank 2, I

Let G be a group with an irreducible spherical BN-pair of rank 2 satisfying the additional condition: (∗) There exists a normal nilpotent subgroup U of B with B=TU, where T=B∩N and |W|≠16 for the Weyl group W=N/B∩N. We show that G corresponds to a Moufang polygon and hence is essentially known.     

Shape-regular polygons in finite planes

The notion of shape in the Gaussian plane was introduced by Lester [5] and extended by Artzy [1]. In this paper we generalize this notion in the affine planesAG(2,q) over the Galois fieldGF(q), q=p ^r andp an odd prime. We investigate the existence of shape-regular polygons and the correspondence between shape-regularity and affine-regularity.Dedicated to the memory of Giuseppe TalliniThe work was supported by the Hungarian National Foundation for Scientific Research, Grant No. F016302 and T017314.     

A note on the solvability of finite Moufang loops

In this short note we show that finite Moufang loops with nilpotent inner mapping groups are solvable. Received: 24 May 2006     

On epimorphisms of spherical Moufang buildings

In this paper, we classify the epimorphisms of irreducible spherical Moufang buildings (of rank $\ge 2$) defined over a field. As an application, we characterize indecomposable epimorphisms of these buildings as those epimorphisms arising from $\mathbb{R}$-buildings.     

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.     

On hyperbolic right-angled polygons

In this paper we find conditions in order to construct hyperbolic right-angledN-gons with the lengths ofN-3 sides given.Explicit formulae for the length of a side in terms of the lengths ofN-3 non-adjacent sides are obtained.Partially supported by CICYT.     

On nilpotent Moufang loops with central associators

In this paper, we investigate Moufang p-loops of nilpotency class at least three for $p>3$. The smallest examples have order $p^5$ and satisfy the following properties: (1) They are of maximal nilpotency class, (2) their associators lie in the center, and (3) they can be constructed using a general form of the semidirect product of a cyclic group and a group of maximal class. We present some results concerning loops with these properties. As an application, we classify proper Moufang loops of order $p^5$, $p>3$, and collect information on their multiplication groups.     

On Linear Representations of Analytic Moufang Loops

Mathematical Notes -     

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).     

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.     

Regular partitions of (weak) finite generalized polygons

In this paper we define a regular m-partition of a distance regular graph as a partition of the vertex set into m classes, such that the number of vertices of a given class adjacent to a fixed vertex of another class (but possibly the same), is independent of the choice of that vertex in this class. Furthermore, we exhibit a technique to determine exact, discrete or bounding values for the intersection numbers of two such regular partitions of a DRG. As an application, we perform a structural investigation on the substructures of finite generalized polygons and, besides some new results, we give unifying, alternative and more elegant proofs of the results in Offer (J Combin Theory Ser A 97: 184–186, 2002) and Offer (Discrete Math 294: 147–160, 2005). The first author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium) (F.W.O.).