Primitive collineation groups of ovals with a fixed point

AbstractWe investigate collineation groups of a finite projective plane of odd order n fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which the group fixes a point off the oval is considered. We prove that it occurs in a Desarguesian plane if and only if (n+1)/2 is an odd prime, the group lying in the normalizer of a Singer cycle of PGL(2,n) in this case. For an arbitrary plane we show that the group cannot contain Baer involutions and derive a number of structural and numerical properties in the case where the group has even order. The existence question for a non-Desarguesian example is addressed but remains unanswered, although such an example cannot have order n≤23 as computer searches carried out with GAP show

Two-transitive ovals

An oval script O sign of a projective plane is called two-transitive if there is a collineation group G fixing script O sign and acting 2-transitively on its points. If the plane has odd order, then the plane is desarguesian and the oval is a conic. In the present paper we prove that if a plane has order a power of two and admits a two-transitive oval, then either the plane is desarguesian and the oval is a conic, or the plane is dual to a Lüneburg plane. © de Gruyter 2006

Abstract hyperovals, partial geometries, and transitive hyperovals

Includes bibliographical references.2015 Summer.A hyperoval is a (q+2)- arc of a projective plane π, of order q with q even. Let G denote the collineation group of π containing a hyperoval Ω. We say that Ω is transitive if for any pair of points x, y is an element of Ω, there exists a g is an element of G fixing Ω setwise such that xg = y. In1987, Billotti and Korchmaros proved that if 4||G|, then either Ω is the regular hyperoval in PG(2,q) for q=2 or 4 or q = 16 and |G||144. In 2005, Sonnino proved that if |G| = 144, then π is desarguesian and Ω is isomorphic to the Lunelli-Sce hyperoval. For our main result, we show that if G is the collineation group of a projective plane containing a transitivehyperoval with 4 ||G|, then |G| = 144 and Ω is isomorphic to the Lunelli-Sce hyperoval. We also show that if A(X) is an abstract hyperoval of order n ≡ 2(mod 4); then |Aut(A(X))| is odd. If A(X) is an abstract hyperoval of order n such that Aut(A(X)) contains two distinct involutions with |FixX(g)| and |FixX(ƒ)| ≥ 4. Then we show that FixX(g) ≠ FixX(ƒ). We also show that there is no hyperoval of order 12 admitting a group whose order is divisible by 11 or 13, by showing that there is no partial geometry pg(6, 10, 5) admitting a group of order 11 or of order 13. Finally, we were able to show that there is no hyperoval in a projective plane of order 12 with a dihedral subgroup of order 14, by showing that that there is no partial geometry pg(7, 12, 6) admitting a dihedral group of order 14. The latter results are achieved by studying abstract hyperovals and their symmetries

On homogeneous planar functions

Let $p$ be an odd prime and \F_q be the finite field with $q=p^n$ elements. A planar function f:\F_q\rightarrow\F_q is called homogenous if $f(\lambda x)=\lambda^df(x)$ for all \lambda\in\F_p and x\in\F_q, where $d$ is some fixed positive integer. We characterize $x^2$ as the unique homogenous planar function over \F_{p^2} up to equivalence.Comment: Introduction modified to: 1. give the correct definition of equivalence, 2. add some references. Other part unaltere

Unitals in projective planes revisited

This thesis revisits the topic of unitals in finite projective planes. A unital U in a projective plane of order q2 is a set of q3 + 1 points, such that every line meets U in one or q + 1 points. Unitals are an important class of point-set in finite projective planes, whose combinatorial and algebraic properties have been the subject of considerable study. In this work, we summarise, revise, and extend contemporary research on unitals. Chapter 1 covers the necessary prerequisites to study unitals and related objects in finite geometry. In Chapter 2, we focus on Buekenhout-Tits unitals and answer some open problems regarding their equivalence, stabilisers and feet. The results presented in Chapter 2 are also available in a preprint paper [22]. Following this, Chapter 3 summarises recent results on Buekenhout- Metz unitals, and presents a small result on the intersection of ovoidal-Buekenhout-Metz unitals and Buekenhout-Metz unitals. Chapter 4 highlights Kestenband arcs and their relationship to Hermitian unitals, and makes explicit a proof of their equivalence. Finally in Chapter 5, we review our understanding of Figueroa planes. Beyond describing ovals and unitals in Figueroa planes, we also suggest generalisations of their constructions to semi-ovals

Relation between o-equivalence and EA-equivalence for Niho bent functions

Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions. However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials. In the present work we study, for a given o-polynomial, a general construction which provides all possible o-equivalent Niho bent functions, and we considerably simplify it to a form which excludes EA-equivalent cases. That is, we identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions derived from o-equivalence of any given Niho bent function. Furthermore, we determine all pairwise EA-inequivalent Niho bent functions arising from all known o-polynomials via o-equivalence.publishedVersio