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 and
where
for all
then A is a semifield if the following holds true:
1. is a group and
2. The identity map
3. If then there exists a unique
such that
, and conversely if
is a vector space over
and
satisfying
and
then
is a semifield.
• If T is the group of translations of the affine plane Π and is the group of elations of the projective plane
with axis
then
1. and
2.
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.