On certain finite geometric structures

Abstract

The purpose of this article is to investigate certain finite geometric structures, in particular semifields over finite fields, spread-sets, translation planes as a special type of affine planes, and projective planes. Quasifields and their normed spread-sets are also considered. Among other results, the following main results are proved.

• If A is a vector semifield a finite field $\mathbb{F}$ and $\mathbb{B}=\left\{f_a\right|0\ne aA\},$ where $f_a(x)=xa$ for all $x\in A,$ then A is a semifield if the following holds true:

1. $(\mathbb{B}\cup \{0\},+)$ is a group and $\mathbb{B}\subseteq GL(A).$

2. The identity map $I\in \mathbb{B}.$

3. If $a,b\in A,a\ne 0,$ then there exists a unique $g\in \mathbb{B}$ such that $a^g=b$, and conversely if $A$ is a vector space over $\mathbb{F}$ and $\mathbb{B}\subseteq GL(A)$ satisfying $1,2$ and $3$ then $A$ is a semifield.

• If T is the group of translations of the affine plane Π and $E_{\infty }(\widehat{\Pi })$ is the group of elations of the projective plane $\widehat{\Pi }$ with axis ${\widehat{L}}_{\infty },$ then

1. $T\le \mathrm{Aut}(\Pi )$ and

2. $E_{\infty }(\widehat{\Pi })\le \mathrm{Aut}{(\widehat{\Pi })}_{{\widehat{L}}_{\infty }}.$

Keywords:

• Autotopism group
• collineation group
• projective planes
• quasifields
• semifield
• shear
• translation planes

2020 Mathematics Subject Classifications:

• 51E15
• 51A40
• 17A35
• 05B25

Acknowledgments

The authors are grateful to Kuwait Foundation for the Advancement of Sciences for supporting this research no. PR18-16SM-02, and to the consultants Abdullah Alazemi, Faris Alazemi and to the referee for their remarks on this article.