Quasifields and Translation Planes of the Smallest Even Order

P. Shtukkert

The Bulletin of Irkutsk State University. Series Mathematics, Tome 7 (2014), pp. 141-159

Résumé

Constructs of different classes of finite non-Desargues translation planes and quasifields closely related. It used by computer calculations since the middle of last century. We study semifields of order 32 and quasifields of order 16 of corresponding translation planes. It is known that translation planes of any order $p^n$ for a prime p can be constructed by using a coordinatizing set $W$ of order $n$ over the field of order $p$. By using a spread set we providing $W$ of structure of quasifield. The plane is set to be a semifield plane if $W$ is a semifield. The plane is Desargues if $W$ is a field. It is well-known that semifield planes are isomorphic if and only if their semifields are isotopic. Structure of quasifields of order $p^n$ has been studied a few, even for small $n$. In 1960 Kleinfeld classified quasifields of order 16 with kernel of order 4 and all semifields of order 16 up to isomorphisms. Later Dempwolf and other completed the classification of all translation planes of order 16 and 32. We construct 5 semifields of order 32 and 7 quasifields of order 16 of non-Desargues planes by using their spread sets. For these semifields and for these quasifields (partially) our main results list for them introduced orders of all non-zero elements and all subfields.

Détail

Keywords: translation planes, spread set, quasifield, semifield, order of element of loop.

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