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Finite nets, numerical invariants

I am studying the following paper: Finite Nets, Numerical Invariants (R. H. Bruck) for a school project about finite geometry. While I understand most of what is written, some points are rapidly ...
tidann's user avatar
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3 votes
1 answer
40 views

Example of a finite projective plane which is not a translation plane

I've been studying finite projective geometry for several weeks and I came across the fact that the most studied planes are the translation planes. Is there any known example (of minimum order if ...
Milan Boutros's user avatar
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Are there interesting finite geometries with angles or triangles which satisfy a Pythagorean-like identity?

I was playing around with Pythagoras's Theorem, and the proofs I was looking at involved areas and congruent triangles. I was wondering how much of this machinery we actually need, and especially if ...
Rebecca J. Stones's user avatar
2 votes
1 answer
45 views

If a finite group acts faithfully and primitively on the points of a configuration, will the figure be flag transitive?

Suppose we have a finite group G, and we had a configuration of points, lines etc. and lets say that the group acts faithfully and primitively on the set of points. Will the group be guaranteed to be ...
Sean Miller's user avatar
1 vote
1 answer
45 views

Maximal set of mutually skew lines in a finite projective space

What is the maximum size of a set of mutually-skew lines in a finite projective space? The total number of lines in a finite projective space is well-known ($(q^2 + 1)(q^2 + q + 1)$) and mentioned in ...
Dale's user avatar
  • 273
0 votes
1 answer
67 views

Are there parameters such that a combinatorial $(n_s,m_t)$ configuration does not exist?

It is well known that given a $(n_s,m_t)$ configuration the following must hold: $$ms=nt$$ $$s(t-1)+1\leq m$$ $$t(s-1)+1\leq n$$ However, for example, a $(43_7,43_7)$ configuration would be an order 6 ...
B. Peet's user avatar
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0 votes
1 answer
73 views

Are there ${n^2+n+1}_{n+1}$ configurations that are not projective planes?

Finite projective planes can be considered as combinatorial ${n^2+n+1}_{n+1}$ configurations. So for example the order 2 projective plane (Fano plane) is a $7_3$ configuration. It is known that the ...
B. Peet's user avatar
  • 109
-1 votes
1 answer
109 views

Does every two-set partition of a projective plane contain a line?

Define a two-set partition of a projective plane as a partition of the points into two sets. Does there exist for any two-set partition a set in the partition that contains a line? What about infinite ...
mathlander's user avatar
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1 vote
0 answers
61 views

Conditions for realizability of finite incidence structures in Euclidean plane

Given a finite incidence structure or equivalently a finite collinearity structure (satisfying standard set of axioms), I am interested in sufficient conditions for realizability in Euclidean plane (...
Kulisty's user avatar
  • 1,478
15 votes
1 answer
343 views

How many ways to arrange $n$ points in $(\Bbb F_q)^2$ with no three collinear?

How many ways are there to arrange $n$ points in the finite field plane $(\Bbb F_q)^2$ with no three of the points collinear? An easy upper bound is $(q^2)^n=q^{2n}$, but of course it's less than that....
Akiva Weinberger's user avatar
3 votes
0 answers
69 views

Common divisor of Gaussian coefficient expressions

I have a question about common divisors of some expressions involving Gaussian coefficients, in particular in the case ${n \brack 1}_{q} = \frac{q^{n}-1}{q-1}$ where $q$ is a prime power. It is well ...
xxxxxxxxx's user avatar
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1 vote
1 answer
187 views

Lower bound of the Turan number $ex(n,C_4)$.

I want to show the lower bound $\Omega(n^{3/2})$ for the Turan number $ex(n,C_4)$. For a prime power $q$, a finite affine plane of order $q$ has $q^2$ points and $q^2 + q$ lines; each line contains $q$...
ensbana's user avatar
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0 votes
0 answers
51 views

A $\bmod p$ version of the Frobenius coin problem.

Let $x_1,\dots,x_d$ be $d$ integers having greatest common divisor equal to $1$. By Bézout's Lemma, there exists a least $\kappa(x_1,\dots,x_d) \in \mathbb{N}$ such that any $k \geq \kappa(x_1,\dots,...
RB1995's user avatar
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1 vote
1 answer
51 views

Finding a bound for the number of cards in a deck

A deck of cards is such that each card has $n$ images drawn inside it and such that each pair of cards has exactly one image in common, but no image is present on all the cards. The question is to ...
Lorenzo Catani's user avatar
2 votes
1 answer
60 views

Does this combinatorical condition characterize the (kernels of) non-degenerate quadratic forms $q:V\to \mathbb{F}_2$?

Let $V$ be an finite-dimensional vector space over $\mathbb{F}_2$ and let $f:{V\choose 2}\to V$ be the addition map defined by $f(\{x,y\})=x+y$. We are looking for subsets $X\subseteq V$ for which ...
Mastrem's user avatar
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3 votes
0 answers
153 views

State of Research on Projective Planes of Order 11

It's been 3 decades since it was shown that there is no projective plane of order 10. I can find lots of commentary and references on the state of play on the attempt to show there is no projective ...
C Monsour's user avatar
  • 8,236
1 vote
1 answer
200 views

Checking if 8 points in the projective plane lie on a singular cubic.

I need to check if 8 points of $\mathbb{P}^2$ (over a finite field) lie on a singular cubic with one of them a double point. I know that to check if a point is singular we suffice to compute the ...
Frankie123's user avatar
0 votes
1 answer
237 views

Non singular degenerate conics in the projective plane

I was trying to count degenerate conics in the projective plane $\mathbb{P}^2(\mathbb{F}_q)$ and I discovered what ooks like a "paradox" (I know it's not but I can't solve it). Given a point ...
Frankie123's user avatar
0 votes
1 answer
73 views

In a finite affine space $(\mathbb{Z}/p)^3$, has at most $p^2 +p +1$ lines passing through a point.

It is a simple question not much backgrond information is required. However, I do need ton understand the proof of why. Prove the finite affine space $(\mathbb{Z}/p)^3$, has only $p^2 +p +1$ lines ...
ben huni's user avatar
  • 173
0 votes
1 answer
34 views

dual of an ovoid in $PG(3,q)$

In $PG(3,q)$, a $(q^2+1)$-cap is an ovoid. A $(q^2+1)$-cap is a set of $q^2+1$ points, no three of which are collinear. What is the dual of an ovoid? I know how to get the dual of a statement ...
mandella's user avatar
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1 vote
1 answer
36 views

if $|\mathcal{H}_1 \cap \mathcal{H}_2|> n+1$ then $\mathcal{H}_1 =\mathcal{H}_2$ for hyperovals of an even order plane

Let $\Pi$ be a projective plane of even order $2n$. Let $\mathcal{H}_1,\mathcal{H}_2$ be two hyperovals in $\Pi$. Show that if $|\mathcal{H}_1 \cap \mathcal{H}_2|> n+1$ then $\mathcal{H}_1 =\...
mandella's user avatar
  • 1,852
1 vote
1 answer
79 views

line at infinity intersected with a line from points of the parabola

Suppose we are looking at a parabola as a conic in $PG(2,\mathbb{K})$ with the line at infinity denoted $\ell_\infty$. I am still working on a previous problem I posted here. Suppose we have two ...
mandella's user avatar
  • 1,852
9 votes
1 answer
417 views

groups of conics

Let $\mathcal{C}$ be a conic on the projective plane $PG(2,\mathbb{F})$ where $\mathrm{char}\mathbb{F}\neq 2$. Let $\ell$ be a line and let $N$ be a point on $\mathcal{C}\setminus \ell$. For $A,B\in \...
mandella's user avatar
  • 1,852
2 votes
0 answers
160 views

What is known about collineation groups in general?

Is there any reference for collineation groups of finite projective planes? By collineation of a projective plane $\pi=(P,L,i)$ I mean a bijective function $f:P\to P$ which preserves collinearity. Are ...
augustoperez's user avatar
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3 votes
0 answers
111 views

Sharply 2-transitive subgroups of the affine group $AGL_d(F)$

I'm attempting exercise 2.8.14 of Dixon & Mortimer. It asks you to show that for $d\geq 1$, $AGL_d(F)$ contains a sharply 2-transitive subgroup $H$. For $d=1$ this is easy, since $AGL_1(F)$ itself ...
mathma's user avatar
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-2 votes
1 answer
391 views

Affine Plane of Order $5$ Picture?

I am unable to construct an affine plane of order $5$. I currently am able to construct an affine plane for orders $3$ and $4$ but I am not able to figure out the construction of an affine plane of ...
Va1or's user avatar
  • 11
1 vote
2 answers
188 views

Show that the harmonic conjugates are collinear - Menelaus and Ceva's Theorems with homogeneous coordinates

Suppose we have a triangle $\triangle ABC$ such that $D$ is an arbitrary point on $BC$, $E$ is an arbitrary point on $AB$ and $F$ is an arbitrary point on $AC$. Let $G$ be the harmonic conjugate of $E$...
mandella's user avatar
  • 1,852
3 votes
1 answer
518 views

About Fano plane symmetries

I have been looking for realizations of order $21$ metacyclic group. I asked about this yesterday and learned some very good information about it Realization of the metacyclic group of order 21 I was ...
user avatar
2 votes
1 answer
318 views

Projective plane - Blocking set relative to line from an oval

Let $\pi=PG(2,n)$ be a desarguesian projective plane of odd order $n$ and $\mathcal{L}$ be a subcollection of lines of $\pi$. Then a set $B$ of points of $\pi$ is called a blocking set relative to $\...
Thomas Lesgourgues's user avatar
0 votes
1 answer
818 views

How many points belong to a line in a finite field?

This is the continuation of this question: What is the size of the set of lines in a finite field $\mathbb{F}_q$ of order $q$, where $q$ is a prime power? Now I need to show that exactly $q$ points ...
ensbana's user avatar
  • 2,287
-1 votes
1 answer
219 views

What is the size of the set of lines in a finite field $\mathbb{F}_q$ of order $q$, where $q$ is a prime power?

Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is a prime power. Show that the set $\mathcal{L} = \{\{(x,y) : ax + by = c\} : a,b,c \in \mathbb{F}_q \text{, and at least one of a,b is ...
ensbana's user avatar
  • 2,287
2 votes
0 answers
43 views

Dual concept to linear error-correcting codes

Let $d$ and $n$ be natural numbers, and let $\mathbb F_q$ be a finite field. For any injection $\sigma: \{1, \dots, d\} \to \{1, \dots, n\}$, we can define a corresponding linear transformation $T_\...
Brent Kerby's user avatar
  • 5,539
1 vote
2 answers
60 views

On some special 5-tuples in projective space $PG(3,2)$

Projective space $PG(3,2)$ has nice 5-tuples of points like $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$, $(1,1,1,1).$ This 5-tuple is "nice" because these points with their pairwise sums cover ...
Alexey Ustinov's user avatar
1 vote
1 answer
61 views

Lower bound on line blocking sets in the finite projective plane

I am looking at Bruen and Rotschild's article on lower bounds for line blocking sets in the finite projective plane of order $q$. In their proof on page 8, they use the summations \begin{align} \sum_{...
Shurik G.'s user avatar
0 votes
2 answers
483 views

Intuition behind lines and points in the projective plane

I've just started learning about projective plane and have trouble understand how to visualize the points and lines in the plane. Specifically, for this lemma below, when its says "all points of $l$ ...
ensbana's user avatar
  • 2,287
3 votes
2 answers
167 views

how many points belong to the quadric $x_0^2+x_1^2+x_2^2+x_3^2=0$ in $\mathbb{P}_3$ over $\mathbb{F}_9$

I have a problem with the following question: how many points belong to the quadric $x_0^2+x_1^2+x_2^2+x_3^2=0$ in $\mathbb{P}_3$ over $\mathbb{F}_9$. How I tried to solve this problem. Here we have ...
user avatar
4 votes
1 answer
254 views

Conjugacy of Singer cyclic groups in $\mathrm{P\Gamma L}$

Motivation This is kind of a follow-up to this question on conjugacy of Singer cyclic groups in GL. The "original" definition of a Singer cycle is not in the GL, but the following slightly different ...
azimut's user avatar
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2 votes
1 answer
425 views

Projective Plane of order n

Let $P=(S,L)$ be a projective plane of order $n$. Let $K$ be a nonempty subset of $S$ with the property that no three points belonging to $K$ are collinear. Prove that $|K|$ is less than or equal to $...
lj_growl's user avatar
  • 317
1 vote
1 answer
1k views

Understanding Witt's Theorem

I just began to learn something about classical polar spaces and now, I'm trying to understand three implications of Witt's theorem. Let $V$ be an $m$-dimensional vector space over a field $K$ ...
user160919's user avatar
1 vote
2 answers
114 views

Binary matrix with fixed inner product.

Suppose that $m,\ n$ are two positive integers such that $m<<n$. Let $a,\ b,\ c$ be the three positive integers such that $a\leq b < c$. Consider a binary matrix $A\in \{0,1\}^{m\times n}$, ...
Shashank Ranjan's user avatar
1 vote
0 answers
39 views

Signed incidence structures

I've been trying to understand GraphQL queries (e.g. Wikidata) with Formal Concept Analysis but my first stumbling block is that most between-object relations at least in the Wikidata ontology are ...
user8948's user avatar
  • 248
1 vote
1 answer
228 views

Projective/ Finite Geometric Basics!

I'm taking intro to coding theory and am having some trouble understanding the basics of Projective Geometry, since our text does not give it much discussion. Namely, if PG(r-1,q) is the set of all ...
Jacob Green's user avatar
4 votes
0 answers
89 views

Translating and inflating a set of $k$-dimensional subspaces of $\mathbb F_p^n$ to form a cover by affine hyperplanes?

Fix a prime number $p$ and consider the affine space $V = \mathbb F_p^n$. Let $k < n/2$. Consider subspaces $V_1, \ldots, V_n \subseteq V$ of dimension $k$, and take $v_i \notin V_i$. Do there ...
Bart Michels's user avatar
  • 26.4k
6 votes
1 answer
2k views

Projective space definitions

My questions are as follows: Are all these different definitions of projective space equivalent? For example, Bezout's theorem holds under all 4 definitions (with an appropriate change in ...
rationalbeing's user avatar
2 votes
2 answers
101 views

no three points in a line on $\mathbb{Z}_p^2$

in finite geometry, the $\mathbb{Z}_p$-plane is $\mathbb{Z}_p^2$, I have proved there are exactly $p+1$ lines pass through a given point in $\mathbb{Z}_p^2$, and by this conclusion there are at most $...
Larry Eppes's user avatar
6 votes
0 answers
605 views

Is a perfect game of Set always possible?

For anyone not familiar with the game of Set, I'll refer you to the description on this question. My question is this: The game ends when there are no more cards remaining in the deck and there are ...
Darrel Hoffman's user avatar
6 votes
2 answers
1k views

Primitive roots of unity occuring as eigenvalues of a product

I am currently trying to understand the proof of Benson's Lemma (1.9.1) in Generalized Quadrangles by Payne and Thas. Background We have two $k × k$ matrices $Q$ and $M$. We want to determine some ...
Santana Afton's user avatar
4 votes
1 answer
144 views

Understanding Generalised Quadrangles

I have a project to do on Generalised Quadrangles, specifically GQ(2,2). The project needs to have information about the construction of GQ(2,2), to prove this construction meets the conditions of a ...
xyz-x's user avatar
  • 523
2 votes
2 answers
945 views

Finite Incidence Geometry Questions

Definition of finite incidence geometry given: Consists of a finite set $P$ of points and a set of nonempty subsets of $P$ called lines, that satisfy the axioms: (F1) Two points determine exactly one ...
Dan P.'s user avatar
  • 283
1 vote
1 answer
347 views

Bound of the size of point set in projective plane.

For those familiar with combinatorial design, a projective plane is a $(q^2+q+1, q+1, 1)$ design. Geometrically, a projective plane of order $q$ is a set of points and lines through these points. ...
xixumei's user avatar
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