Alcune condizioni per la costruzione di un piano finito di traslazione

Basile, Alessandro ; Brutti, Paolo

Bollettino della Unione matematica italiana, Série 8, 4B (2001) no. 2, pp. 429-439

Résumé

The construction of any finite translation plane depends on the appropriate determination of a partition $P(E)$ of a Galois field $E$, together with a set $A$ of automorphisms of $E$ as a vector space. In this paper we obtain sufficient conditions on $P(E)$ and $A$, so that a translation plane is produced. They are also necessary conditions when $|A|=1$. Particularly, we examine the case where $E$ is a two-dimensional vector space. We prove that no translation planes are constructible by a single automorphism, other than the André planes. Furthermore, if $|A|>1$, the conditions previously obtained cannot be satisfied.