Nuclear fusion in finite semifield planes

Abstract

We study the subplanes of finite semifield planes that are coordinatizable by subfields F of some semifield D such that F lies in at least two of the three seminuclear fields N_ℓ(D), N_m(D), and N_r(D). Our main results determine completely the combinatorial configurations associated with such subplanes, and enables, for example, a computational method to determine the number of nuclear planes of order q in semifield planes of order q^t.

The results have a number of applications. Firstly, they imply ‘fusion’ theorems. The most basic one is that if two or more seminuclear subfields, of a semifield D coordinatizing a translation plane π, are each isomorphic to GF(q), then π may be recoordinatized by a semifield E such that the indicated seminuclear fields, of order q, coincide. The most important case is nuclear fusion: if all three seminuclei of D are isomorphic to GF(q) then the nucleus N(E) ≅ GF(q).

Further applications are concerned with semifield spreads π of order q². We classify all such π that admit three homology groups of order q – 1 with dierent axis (the shears axis, the infinite line, and any other component), thus generalizing a theorem of D. E. Knuth, who proved an algebraic version of the result.