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Affinely regular polygons in an affine plane
Full text linkIn this paper we survey results about affinely regular polygons. First, the definitions and classification of affinely regular polygons are given. Then the theory of Bachmann-Schmidt is outlined. There are several classical theorems about regular polygons, some of them having analogues in finite planes, such as the Napoleon-Barlotti theorem. Such analogues, variants of classical theorems are also collected. Affinely regular polygons occur in many combinatorial problems for sets in a finite plane. Some of these results about sharply focused arcs, internal and external nuclei, complete arcs are collected. Finally, bounds on the number of chords of an affinely regular polygon through a point are discussed
Note on disjoint blocking sets in Galois planes
No full textA blocking set in a projective or affine plane is a set of points, which intersects every
line. Blocking sets are particular cases of 1-covers in hypergraphs. For projective planes, the smallest blocking sets are just the lines. Blocking sets containing a line will be called trivial. A blocking set is said to be
minimal (or irreducible) when no proper subset of it is a blocking set. In this paper, we show that there are at least cq disjoint blocking sets in PG(2; q),
where c is about 1/3. The result also extends to some non-Desarguesian planes of order q
Embedding of Classical Polar Unitals in PG(2,q^2)
Full text linkA unital, that is, a block-design 2 - (q^3 + 1, q + 1,1), is embedded in a projective plane II of order q(2) if its points and blocks are points and lines of H. A unital embedded in PG(2, q(2)) is Hermitian if its points and blocks are the absolute points and non-absolute lines of a unitary polarity of PG(2, q(2)). A classical polar unital is a unital isomorphic, as a block-design, to a Hermitian unital. We prove that there exists only one embedding of the classical polar unital in PG(2, q(2)), namely the Hermitian unital. (C) 2017 Elsevier Inc. All rights reserved
Some Multiply Derived Translation Planes with SL(2,5) as an Inherited Collineation Group in the Translation Complement
No full textFinite translation planes having a collineation group isomorphic to SL(2,5) occur in many investigations on minimal normal non-solvable subgroups of linear translation complements. In this paper, we are looking for multiply derived translation planes of the desarguesian plane which have an inherited linear collineation group isomorphic to SL(2,5). The Hall plane and some of the planes discovered by Prohaska are translation planes of this kind of order q^2, provided that q is odd and either q^2 is congruent 1 mod 5 or q is a power of 5. In this paper the case q congruent -1 mod 5 is considered and some examples are constructed under the further hypotesis that q is congruent 2 mod 3, or q is congruent 1 mod 3 and q is congruent 1 mod 4, or q is congruent -1 mod 4, 3 does not divide q and q is congruent 3, 5, or 6 mod 7. One might expect that examples exist for each odd prime power q. But this is not always true according to Theorem 2
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