Some Multiply Derived Translation Planes with SL(2,5) as an Inherited Collineation Group in the Translation Complement

Abstract

Finite translation planes having a collineation group isomorphic to SL(2,5) occur in many investigations on minimal normal non-solvable subgroups of linear translation complements. In this paper, we are looking for multiply derived translation planes of the desarguesian plane which have an inherited linear collineation group isomorphic to SL(2,5). The Hall plane and some of the planes discovered by Prohaska [10], see also [1], are translation planes of this kind of order q ²;, provided that q is odd and either q ²; ≡ 1 mod 5 or q is a power of 5. In this paper the case q ² ≡ -1 mod 5 is considered and some examples are constructed under the further hypothesis that either q ≡ 2 mod 3, or q ≡ 1 mod 3 and q ≡ 1 mod 4, or q ≡ -1 mod 4, 3 ∤ q and q ≡ 3,5 or 6 mod 7. One might expect that examples exist for each odd prime power q. But this is not always true according to Theorem 2.