Vol. 36, No. 3, 1971

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Transition planes constructed from semifield planes

Norman Lloyd Johnson

Vol. 36 (1971), No. 3, 701–711

Let π be an affine plane of order q2 that is coordinatized by a “derivable” semifield 𝒮 = (𝒮,+,). If (𝒮,+) is a right vector space over F = GF(q) then a plane πmay be constructed from π using Ostrom’s method of “derivation.”

The purpose of this article is to examine the planes πand their coordinate structures (𝒮,+,). It is shown, in particular, that (𝒮,+,) is a (right) quasifield which is neither a nearfield nor a semifield. Furthermore, it is shown that πis always of Lenz-Barlotti class IVa. 1.

The automorphism groups of semifields of square order are also briefly investigated.

Mathematical Subject Classification
Primary: 50.70
Received: 3 June 1970
Published: 1 March 1971
Norman Lloyd Johnson