Direct sum decomposition of geometries based on maximal subgroups

In 1961 J. Tits described a way to define a geometry from a group and a collection of subgroups. Such incidence geometries are now studied by the team of F. Buekenhout in Brussels. Here we present theorems about decomposition of PRI geometries into direct sums and we find the full direct sum decomposition of PRI geometries on solvable groups.     

Constructions of rank five geometries for the Mathieu group M 22

We construct nine rank five incidence geometries that are firm and residually connected and on which the Mathieu group M²² acts flag-transitively. The constructions use mainly objects arising from the Steiner systemS(3, 6, 22). One of these geometries was constructed by Meixner and Pasini in [10]. Three of them are obtained from the geometry of Meixner and Pasini using doubling (see [8] or [12]) or similar constructions. The remaining five are new and four of them have a star diagram. These latter four geometries are constructed using special partitions of the 22 points of the Steiner system S(3, 6, 22).     

Maps between buildings that preserve a given Weyl distance

Let Δ and Δ′ be two buildings of the same type (W, S), viewed as sets of chambers endowed with“distance” functions δ and δ′, respectively, admitting values in the common Weyl group W, which is a Coxeter group with standard generating set S. For a given element ω ε W, we study surjective maps ? : Δ → Δ′ with the property that δ(C, D) = ω if and only if Δ′ (?(C), ?(D)) = ω. The result is that the restrictions of ? to all residues of certain spherical types—determined by ω—are isomorphisms. We show with counterexamples that this result is optimal. We also demonstrate that, in many cases, this is enough to conclude that ? is an isomorphism. In particular, ? is an isomorphism if Δ and Δ′ are 2-spherical and every reduced expression of ω involves all elements of S.     

On a problem on chamber systems

It is well known that a geometry belonging to a disconnected diagram is the direct sum of geometries corresponding to the connected components of the diagram. On the other hand, chamber systems with a disconnected diagram exist which do not split as direct products of components of smaller rank. Many finite examples of this kind are discussed in Groups of Lie Type and their Geometries (CUP, 1995, pp. 185–214), but none of them is simply connected. In this article, we construct a simply connected finite example.     

Designs and partial geometries over finite fields

Let be a finite field, and let (, B) be a nontrivial 2-(n, k, 1)-design over . Then each point induces a (k–1)-spread S on /. (, B) is said to be a geometric design if S is a geometric spread on / for each . In this paper, we prove that there are no geometric designs over any finite field .Research partially supported by NSF grant DMS-8703229.     

On homomorphisms between generalized polygons

We consider homomorphisms between abstract, topological, and smooth generalized polygons. It is shown that a continuous homomorphism is either injective or locally constant. A continuous homomorphism between smooth generalized polygons is always a smooth embedding. We apply this result to isoparametric submanifolds.Dedicated to Prof. Dr. H. R. Salzmann on the occasion of his 65th anniversary     

Extended generalized octagons and the groupHe

Let be an extended generalized octagon such that the points of a triple {u, v, w} not on a block are pairwise adjacent if and only if the distance betweenv andw in the local generalized octagon _u equals 3 and there is a thick line through any point of _u . Then is one of the two examples related to the groups 2·L ₃(4).2² andHe. It is also shown that does not admit further extensions.     

Note on the existence of large minimal blocking sets in galois planes

A subsetS of a finite projective plane of orderq is called a blocking set ifS meets every line but contains no line. For the size of an inclusion-minimal blocking setq+ +Sq +1 holds ([6]). Ifq is a square, then inPG(2,q) there are minimal blocking sets with cardinalityq +1. Ifq is not a square, then the various constructions known to the author yield minimal blocking sets with less than 3q points. In the present note we show that inPG(2,q),q1 (mod 4) there are minimal blocking sets having more thanqlog₂ q/2 points. The blocking sets constructed in this note contain the union ofk conics, whereklog₂ q/2. A slight modification of the construction works forq3 (mod 4) and gives the existence of minimal blocking sets of sizecqlog₂ q for some constantc.As a by-product we construct minimal blocking sets of cardinalityq +1, i.e. unitals, in Galois planes of square order. Since these unitals can be obtained as the union of parabolas, they are not classical.     

On stable geometries

Within the concept of projective lattice geometry we are considering the class of stable geometries which have also been introduced in [14]. The investigation of their basic properties will result in fundamental structure theorems which especially give a lattice-geometric characterization of free left modules of rank 6 over proper right Bezout rings of stable rank 2. This yields a proper generalization of previous results of ours.     

Families of polytopal digraphs that do not satisfy the shelling property

A polytopal digraph $G(P)$ is an orientation of the skeleton of a convex polytope P. The possible non-degenerate pivot operations of the simplex method in solving a linear program over P can be represented as a special polytopal digraph known as an LP digraph. Presently there is no general characterization of which polytopal digraphs are LP digraphs, although four necessary properties are known: acyclicity, unique sink orientation (USO), the Holt–Klee property and the shelling property. The shelling property was introduced by Avis and Moriyama (2009), where two examples are given in $d=4$ dimensions of polytopal digraphs satisfying the first three properties but not the shelling property. The smaller of these examples has $n=7$ vertices. Avis, Miyata and Moriyama (2009) constructed for each $d?4$ and $n?d+2$, a d-polytope P with n vertices which has a polytopal digraph which is an acyclic USO that satisfies the Holt–Klee property, but does not satisfy the shelling property. The construction was based on a minimal such example, which has $d=4$ and $n=6$. In this paper we explore the shelling condition further. First we give an apparently stronger definition of the shelling property, which we then prove is equivalent to the original definition. Using this stronger condition we are able to give a more general construction of such families. In particular, we show that given any 4-dimensional polytope P with $n_0$ vertices whose unique sink is simple, we can extend P for any $d?4$ and $n?n_0+d?4$ to a d-polytope with these properties that has n vertices. Finally we investigate the strength of the shelling condition for d-crosspolytopes, for which Develin (2004) has given a complete characterization of LP orientations.     

Algebraic radicals and incidence algebras

In this note we prove the uniqueness of U in a group G with a spherical split-BN-pair of rank ,i.e., if G has such a BN-pair with a nilpotent normal subgroup of B, and , then and is a normal subgroup of G. Here is the corresponding group of Lie-type and the subgroup of generated by all root-subgroups corresponding to positive roots. Received: 19 May 2003     

A characterization of 3-local geometry of M(24)

The largest Fischer 3-transposition group M(24) acts flag-transitively on a 3-local incidence geometry (M(24)) which is a c-extension of the dual polar space associated with the group O ₇(3). The action of the simple commutator subgroup M(24) is still flag-transitive. We show that (M(24)) is characterized by its diagram under the flag-transitivity assumption. The result implies in particular that (M(24)) is simply connected. The geometry (M(24)) appears as a subgeometry in the Buekenhout-Fischer 3-local geometry (F ₁) of the Monster group. The simple connectedness of (M(24)) has played a crucial role in the characterization of (F ₁), which has been achieved recently. When determining the possible structure of the parabolic subgroups we have used an unpublished pushing-up result by U. Meierfrankenfeld.Dedicated to Professor B. Fischer on the occasion of his sixtieth birthday     

Tubes of even order and flat π.C 2 geometries

Atube of even orderq=2 ^d is a setT={L, } ofq+3 pairwise skew lines in PG(3,q) such that every plane onL meets the lines of in a hyperoval. Thequadric tube is obtained as follows. Take a hyperbolic quadricQ=Q ₃ ⁺ (q) in PG(3,q); letL be an exterior line, and let consist of the polar line ofL together with a regulus onQ.In this paper we show the existence of tubes of even order other than the quadric one, and we prove that the subgroup of PL(4,q) fixing a tube {L, } cannot act transitively on . As pointed out by a construction due to Pasini, this implies new results for the existence of flat .C ₂ geometries whoseC ₂-residues are nonclassical generalized quadrangles different from nets. We also give the results of some computations on the existence and uniqueness of tubes in PG(3,q) for smallq. Further, we define tubes for oddq (replacing hyperoval by conic in the definition), and consider briefly a related extremal problem.Dedicated to luigi antonio rosati on the occasion of his 70th birthday     

Wrapping polygons in polygons

This paper is developed toI ₂(2g).c-geometries, namely, point-line-plane structures where planes are generalized 2g-gons with exactly two lines on every point and any two intersecting lines belong to a unique plane.I ₂(2g).c-geometries appear in several contexts, sometimes in connection with sporadic simple groups. Many of them are homomorphic images of truncations of geometries belonging to Coxeter diagrams. TheI ₂(2g).c-geometries obtained in this way may be regarded as the standard ones. We characterize them in this paper. For everyI ₂(2g).c-geometry , we define a numberw(), which counts the number of times we need to walk around a 2g-gon contained in a plane of , building up a wall of planes around it, before closing the wall. We prove thatw()=1 if and only if is standard and we apply that result to a number of special cases.     

On the intersection property of Dubrovin valuation rings

It is shown that of the three axioms Gräter specified for his intersection property of Dubrovin valuation rings in central-simple algebras, the second and third axioms actually follow from the first.

     

On the sharpness of a theorem of B. segre

The theorem of B. Segre mentioned in the title states that a complete arc of PG(2,q),q even which is not a hyperoval consists of at mostq−√q+1 points. In the first part of our paper we prove this theorem to be sharp forq=s ² by constructing completeq−√q+1-arcs. Our construction is based on the cyclic partition of PG(2,q) into disjoint Baer-subplanes. (See Bruck [1]). In his paper [5] Kestenband constructed a class of (q−√q+1)-arcs but he did not prove their completeness. In the second part of our paper we discuss the connections between Kestenband’s and our constructions. We prove that these constructions result in isomorphic (q−√q+1)-arcs. The proof of this isomorphism is based on the existence of a traceorthogonal normal basis in GF(q ³) over GF(q), and on a representation of GF(q)³ in GF(q ³)³ indicated in Jamison [4].     

The translation planes of order 81 that admit SL(2,5), generated by affine elations

The translation planes of order 81 admitting SL(2, 5), generated by affine elations, are completely determined. There are seven mutually non-isomorphic translation planes, of which five are new. Each of these planes may be derived producing another set of seven mutually non-isomorphic translation planes admitting SL(2, 5), where the 3-elements are Baer. Of this latter set, five planes are new.     

On monoids over which all generators satisfy a flatness property

This paper addresses conditions under which all generators in the category of right S-acts (where S is a monoid) satisfy a flatness property. There are characterizations for monoids over which all generators satisfy a flatness property α where α can stand for freeness, projectivity, strong flatness, Condition (P), principal weak flatness and torsion freeness. To our knowledge, the problem has not been studied for other flatness properties such as weak flatness, Condition (E) and regularity. The present paper addresses this gap.