3 edition of **Baer subplanes of projective planes** found in the catalog.

Baer subplanes of projective planes

Rilla Nichols

- 349 Want to read
- 26 Currently reading

Published
**1970**
by Dept. of Mathematics, Carleton University in Ottawa
.

Written in English

- Projective planes.

**Edition Notes**

Bibliography: leaves 92-93.

Statement | by Rilla Nichols. |

Series | Carleton mathematical series ;, no. 124 |

Classifications | |
---|---|

LC Classifications | QA471 .N5 |

The Physical Object | |

Pagination | 93 leaves : |

Number of Pages | 93 |

ID Numbers | |

Open Library | OL4233228M |

LC Control Number | 80513721 |

This work confronts the question of geometric processes of derivation, specifically the derivation of affine planes - keying in on construction techniques and types of transformations in which lines of a newly-created plane can be understood as subplanes of the original plane. The book provides a theory of subplane covered nets without restriction to the finite case or imposing commutativity. In geometry, smooth projective planes are special projective most prominent example of a smooth projective plane is the real projective geometric operations of joining two distinct points by a line and of intersecting two lines in a point are not only continuous but even smooth (infinitely differentiable = ∞).Similarly, the classical planes over the complex numbers, the.

[10, 17] The union of the point set of two disjoint Baer subplane of a Baer subplane from a Desarguesian plane is a saturating set, of size 2 p q+ 2 4 p q+ 2. (The existence of Baer-subplanes in Baer subplanes of PG(2;q) tacitly implies that qis a 4th power.) However, in the general case when the plane is not necessarily Desarguesian or the order. Including numerous worked examples and exercises throughout, the book covers axiomatic geometry, field planes and PG(r, F), coordinating a projective plane, non-Desarguesian planes, conics and quadrics in PG(3, F). Assuming familiarity with linear algebra, elementary group theory, partial differentiation and finite fields, as well as some.

The construction of the plane is the standard construction based on a quasifield (see Quasifield#Projective planes for the details.). To build a Hall quasifield, start with a Galois field, F = GF (p n) {\displaystyle F=\operatorname {GF} (p^{n})} for p a prime and a quadratic irreducible polynomial f (x) = x 2 − r x − s. For example, unitals play a parallel role to Baer s- planes when considering extreme values for the size of a blocking set in a square order projective plane (see Section ). Moreover, unitals meet the upper bound for the number of absolute points of any polarity in a square order projective plane .

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Baer subplanes of projective planes (Carleton mathematical series) Unknown Binding – January 1, by Rilla Nichols (Author) See all formats and editions Hide other formats and editions. Enter your mobile number or email address below and we'll send you a link to download the free Kindle App.

Author: Rilla Nichols. This paper is an addition to the book [54] on Compact projective planes. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical. Abstract. Baer subplanes are subplanes of order q of a projective plane of order q intersection configurations are well known.

The concept of Baer subplanes is extended to n dimensions and two dimensional results are generalised to Baer subspaces of PG(n,q 2).Cited by: In PG(2,q2) let ℓ∞ denote a fixed line, then the Baer subplanes which intersect ℓ∞ in q+1 points are called affine Baer subplanes. Call a Baer subplane of PG(2,q2) non-affine if it.

The projective plane PG(2;p) de ned over the nite eld GF(p) where pis prime, does not have non-degenerate subplanes.

On the other hand, for any square prime power q, PG(2;q) contains non-degenerate subplanes of order p q, these are called Baer subplanes. Moreover, one can partition the point set of the plane into the point sets of q p q+ 1.

The real plane is a Baer subplane of the complex projective plane in the following sense: each complex line contains a real point and, dually, each complex point is on a real line. In the first part of this survey main aspects of the general theory will be reviewed.

The second part is concerned with finite planes. Theorem I. 1 is proved in Section 3. At the end of this paper we discuss some related problems and open questions.

Baer partitions and Kestenband arcs Proposition Let Bo and B1 be two Baer subplanes of the desarguesian projective plane P = PG(2,q2), and let To and zl be the Baer involutions defined by Bo and BI, respectively.

Whenever II is a Desarguesian plane of square order q X, we can construct such a 3-blocking set since the plane has a partition into Baer subplanes [13]. This raises the question of whether a set with the parameters (i.e., size and intersection numbers) of the union of three mutually disjoint Baer subplanes does split into three Baer subplanes.

We show that, for small t, the smallest set that blocks the long secants of the union of t pairwise disjoint Baer subplanes in. Conics in Baer subplanes. Susan G. Barwick, Wen-Ai Jackson, and Peter Wild Full-text: Access denied (no subscription detected) The main result is to show that a conic in a tangent Baer subplane of PG (2, q 2) 51E Combinatorial structures in finite projective spaces.

Comments. A projective plane is called Desarguesian if the Desargues assumption holds in it (i.e. if it is isomorphic to a projective plane over a skew-field). The idea of finite projective planes (and spaces) was introduced by K. von Staudt, pp.

87– The fact that a finite projective plane with doubly-transitively acting group of collineations is Desarguesian is the Ostrom–Wagner. Definition. A proper subplane of a projective plane is called a Baer subplane if the following two conditions are true.

Given any point there exists exactly one line through p. Given any line there exists exactly one point on l. Of course, Baer subplanes may be characterized in terms of coordinatizing ternary fields. Lemma. Let be a subplane of a projective plane.

Subplanes. A subplane of a projective plane is a subset of the points of the plane which themselves form a projective plane with the same incidence relations. (Bruck ) proves the following theorem. Let Π be a finite projective plane of order N with a proper subplane Π.

This book is a monograph on unitals embedded in?nite projective planes. For example, unitals play a parallel role to Baer s- planes when considering extreme values for the size of a blocking set in a square order projective plane (see Section ).

We introduce and study embeddings of graphs in finite projective planes, and present related results for some families of graphs including complete graphs and complete bipartite graphs.

We also make connections between embeddings of graphs and the existence of certain substructures in a plane, such as Baer subplanes and arcs. Conics in secant Baer subplanes In this section we consider the Bruck-Bose representation of F q-conics in secant Baer subplanes of PG(2;q2), in particular looking at the relationship with the lines of the hyperbolic congruence of g;gq.

Theorem Let Cbe an F q-conic in a Baer subplane B secant to ‘ 1. The F q2-conic C meets ‘. Together with the lines containing at least 2 of them they form a projective plane of order 2 called a Baer-subplane. As above, the we can count the number of Baer subplanes. There are of them, forming one orbit under the group, but falling apart in 3 orbits of size under the action of.

Summerizing we obtain the following. Subplanes of projective planes of order Caliskan, Cafer; Magliveras, Spyros We determine orbit representatives of all proper subplanes generated by quadrangles of a Veblen-Wedderburn (VW) plane Π of order 11 2 and the Hughes plane Σ of order 11 2 under their full collineation groups.

In Π, there are 13 orbits of Baer. Collineations on Baer Subplanes. Let P be a projective plane of order n 2. A subplane B of order n of P is called Baer subplane. Baer suplanes are exactly the maximal subplanes of P. InducedCollineation(baerplane, baercoll, point, image, planedata, embedding) O.

Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share. projective planes; in both topics we provide some new results. Keywords: minimal blocking set, Baer subplane, stabiliser of a Baer subplane, Hall plane, Andr e plane, double blocking set, value set of polynomials.

1 Introduction In nite geometry one often studies combinatorial analogues of classical substructures of Galois geometries.Baer subplanes all of which are desarguesian, 2 orbits of subplanes of order 3 and at most ; distinct Fano subplanes.

This work was motivated by the well known question: \Does there exist a non-desarguesian projective plane of prime order p?". The question remains unsettled. 1 Introduction.Baer subplanes of projective planes are extended Kirkman systems with v = q and m = 1.

We have-found no further examples with v = qs The smallest Kirkman systems with k > 2 and m > 1 are the Kirkman systems with the classical parameters v = 15, b = 35, r 7, k 3, m 2.

In Sections 3 and 5 of this paper, we investigate the following assumption.