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):

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].

- 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 defined as in 2.1.2 is of the form with the additive group of the near-ring, the fixed-point-free automorphism group used to obtain the near-ring multiplication, and the normalizer of in the automorphism group of .

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 can always be expressed as a semi-direct product of and a subgroup of including the normalizer of . A positive answer to this question was given under certain additional conditions on the design [KK94].
- characterize which Frobenius groups give isomorphic designs.

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

- [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 -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.