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Articles

On certain finite geometric structures

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Pages 3734-3743 | Received 29 Jun 2021, Accepted 11 Feb 2022, Published online: 03 Mar 2022
 

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 F and B={fa|0aA}, where fa(x)=xa for all xA, then A is a semifield if the following holds true:

1. (B{0},+) is a group and BGL(A).

2. The identity map IB.

3. If a,bA,a0, then there exists a unique gB such that ag=b, and conversely if A is a vector space over F and BGL(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(Π̂) is the group of elations of the projective plane Π̂ with axis L̂, then

1. TAut(Π) and

2. E(Π̂)Aut(Π̂)L̂.

2020 Mathematics Subject Classifications:

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.

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