# Semifield planes

### Related pages

Translation planes
Lenz classification
Semifields
Isotopism
A semifield plane is a projective plane P(K) defined over a semifield K. Given such a plane P(K), the line is a translation axis, and the point is a dual translation axis, respectively. In particular, P(K) is of Lenz type at least V. In fact, the Lenz type is exactly V, unless the semifield K is an alternative field.

Let be the stabilizer

Unless K is an alternative field, we have in fact The group contains the translation group as an invariant subgroup

Moreover, our plane admits the following shears

and we get an invariant subgroup

An autotopism triangle in P(K) is a triangle of the form with The group acts on the set of all autotopism triangles, and is a regular, normal subgroup of this action. In particular, the group is a semidirect product where denotes the stabilizer

The group is canonically isomorphic to the autotopism group of our semifield K.

For finite semifield planes, there is the following conjecture.

Conjecture. The automorphism group of a finite, non-desarguesian semifield plane is solvable.

Since a finite alternative field is already a field, it is easily seen that the conjecture is equivalent with the following algebraic version.

Conjecture. (Algebraic version) If K is a finite semifield but not a field, then the autotopism group of K is solvable.