Related pagesTranslation planes
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.