Flag-transitive L_h.L*-geometries

Bulletin of the American Mathematical Society, 1994

We announce the classification of two related classes of flag-transitive geometries. There is an infinite family of such geometries, related to the nonsplit extensions 3' 2h • Sp2n(2), and twelve sporadic examples coming from the simple groups Af22 > ^23 , .A/24. He , Co\ , C02 , J4 , BM , M and the nonsplit extensions 3 • A/22 , 323 ■ C02 , and 34371 • BM.

European Journal of Combinatorics, 1995

We consider geometries belonging to the following diagram of rank n >~ 4,

Discrete Mathematics, 1993

Pasini, A., Flag-transitive C,-geometries, Discrete Mathematics 117 (1993) 169-182.

1998

A C2:L-geometry is a geometry of rank 3 with elements called points, lines and quads, where residues of points are linear spaces, residues of lines are generalized digons and residues of quads are generalized quadrangles. Some sufficient conditions can be found in the literature for a C2:L-geometry to be a quotient of a truncated Cn-building. We shall weaken those conditions in this paper.

Geom Dedic, 1993

We consider tilde-geometries (or T-geometries), which are geometries belonging to diagrams of the following shape:

Stichting Mathematisch Centrum. Zuivere Wiskunde, 1982

The pulrpose of this note is to give an explicit construction of a finite flag-transitive GAB with an extended G 2-diagram having the group G 2 (3) as automorphism group. In our example it will be apparent that the intersection property is satisfied.

Let H be a geometric hyperplane of a classical finite generalized quadrangle Q and let C = Q \ H be its complement in Q, viewed as a point-line geometry. We shall prove that C admits a flag-transitive automorphism group if and only if H spans a hyperplane of the projective space in which Q is naturally embedded (but with Q viewed as Q(4, q) when Q = W (q), q even).

2003

Abstract. Until the 1990's the only known finite linear spaces admitting line-transitive, pointimprimitive groups of automorphisms were Desarguesian projective planes and two linear spaces with 91 points and line size 6. In 1992 a new family of 467 such spaces was constructed, all having 729 points and line size 8. These were shown to be the only linear spaces attaining an upper bound of Delandtsheer and Doyen on the number of points.

with thick lines are got from thick buildings of type D, or D4 in standard ways ($57.12 and

Beitrage zur Algebra und Geometrie

The classification of finite flag-transitive linear spaces is almost com- plete. For the thick case, this result was announced by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl, and in the thin case (where the lines have 2 points), it amounts to the classification of 2-transitive groups, which is generally considered to follow from the classification of finite simple groups. These two clas- sifications actually leave an open case, which is the so-called 1-dimensional case. In this paper, we work with two additional assumptions. These two conditions, namely (2T)1 and RWPri, are taken from another field of study in Incidence Ge- ometry and allow us to obtain a complete classification, which we present at the end of this paper. In particular, for the 1-dimensional case, we show that the only (2T)1 flag-transitive linear spaces are AG(2,2) and AG(2,4), with A L(1,4) and A L(1,16) as respective automorphism groups.

Bulletin of The Australian Mathematical Society, 2007

A method of embedding nk configurations into projective space of k-1 dimensions is given. It breaks into the easy problem of finding a rooted spanning tree of the associated Levi graph. Also it is shown how to obtain a "complementary" n n-k "theorem" about projective space (over a field or skew-field F) from any n* theorem over F. Some elementary matroid theory is used, but with an explanation suitable for most people. Various examples are mentioned, including the planar configurations: Fano 73, Pappus 93, Desargues IO3 (also in 3d-space), Mobius 84 (in 3d-space), and the resulting 7t in 3d-space, 96 in 5d-space, and IO7 in 6d-space. (The Mobius configuration is self-complementary.) There are some n/t configurations that are not embeddable in certain projective spaces, and these will be taken to similarly not embeddable configurations by complementation. Finally, there is a list of open questions.

CAUCHY, 2016

In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system S(t, k, v) is a set of v points and k blocks which satisfy that every t-subset of v-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with t = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order. In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.

数理解析研究所講究録, 1999

Discrete Mathematics, 1986

We consider geometries belonging to the following diagrams: 0 1 2 3 (F,): o o o o, we say that the elements of type 0 are points, the elements of type 1 are lines, the elements of type 2 are planes. If F belongs to C. and n I> 4, then the elements of type n-1 are called hyperlines. The hyperlines of a geometry in F4 are the elements of type 3. If the geometry belongs to C, and n i> 5, then the elements of type n -2 are called subspaces of codimension 1. We often use sentences such as "the point a lies on the line b", "the line b passes through the point a", "the 0012-365X/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland) 46 A. Pasini point a and the line b belong to the plane c", and so on. Their meaning is Proposition 2 (Tits [10, Theorem 1]). All buildings are 2-connected. The role of Properties (LL), (LH), (HH) and (O) is made clear by the following proposition, which collects contributions by Tits, Timmesfeld, Brouwer and Cohen.

CAUCHY: Jurnal Matematika Murni dan Aplikasi, 2016

In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system S(t, k, v) is a set of v points and k blocks which satisfy that every t-subset of v-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with t = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.