2.2 Goals of the project

Based on the existing results described in the list 1 to 6 of Section 2.1 we want to attack the following problems (The numbering corresponds to that of 2.1):

1. In the part of the project that deals with Frobenius groups, we want to

   • explicitly state the irreducible representations of fixed-point-free automorphism groups over finite fields.
   • characterize classes of Frobenius groups with non-abelian kernel beyond the construction of individual examples as in [KW91,Aic01].

2. There exists a great variety of established methods to obtain designs (see e.g. [BJL99b,BJL99a]). It is therefore necessary for us to
   • compare the classical constructions with the construction via planar near-rings (Frobenius groups) in order to find out which approach is more efficient;
   • determine which of the BIB-designs from planar near-rings are new.

   In [Mod89] it is conjectured that the automorphism group of a design $D$ defined as in 2.1.2 is of the form $G\cdot N_A(\Phi)$ with $G$ the additive group of the near-ring, $\Phi$ the fixed-point-free automorphism group used to obtain the near-ring multiplication, and $N_A(\Phi)$ the normalizer of $\Phi$ in the automorphism group $A$ of $G$.

   This conjecture in its full generality could already be refuted using SONATA. We now want to

   • determine for which isomorphism types of the corresponding Frobenius group (in particular of the Frobenius complement) Modisett's conjecture is valid;
   • determine whether the automorphism group of $D$ can always be expressed as a semi-direct product of $G$ and a subgroup of $A$ including the normalizer of $\Phi$. A positive answer to this question was given under certain additional conditions on the design [KK94].
   • characterize which Frobenius groups give isomorphic designs.

3. [4.] Structure theory of planar near-rings and $1$-primitive near-rings. We want to:
   • find the connection between the structure theorems on $1$-primitive near-rings and those on planar near-rings;
   • generalize the description of $1$-primitive near-rings to $1$-primitive composition algebras, and find generalizations of planar near-rings that also yield geometric structures;
   • characterize the $p$-groups $G$ and fixed-point-free automorphism groups $\Phi$ on $G$ such that the sub-near-ring of $A(G)$ generated by the elements of $\Phi$ by pointwise addition and composition is local.

4. [5.] In order to construct and count the polynomial functions on Frobenius groups using the results in [Aic01], we have to determine the restrictions of polynomial functions to the Frobenius kernel.

   We want to

   • determine the polynomial functions of Frobenius group by using the results in [Aic01];
   • determine $n$-ary polynomial functions on Frobenius groups;
   • find the compatible functions on Frobenius groups;
   • characterize the affine complete Frobenius groups, that is Frobenius groups for which every compatible function is polynomial.