2.1.4 Positioning of this project

In this project (a joint project with the Taiwanese National Science Council), researchers from Tainan (Taiwan), Hamburg, and Linz will work based on their previous research in the following relevant areas:

1. Construction of Frobenius groups: For applications like building designs with specific parameters, it is necessary to provide Frobenius groups of specified size and isomorphism type (cf. [Boy01]). [KK95] presents a characterization of Frobenius groups where both the kernel and the complement are abelian. In general, Frobenius groups with abelian kernel can be obtained via representations of fixed-point-free automorphism groups over finite fields [May98,May00]. These representations can be found using [Bro01].

   For the construction of Frobenius groups with non-abelian kernel, there is no systematic approach known. Examples of such groups have been described, e.g., in [KW91,Aic01].

2. Designs from planar near-rings: Various ways to obtain designs and even BIB-designs from planar near-rings (or equivalently from Frobenius groups) are presented in [Cla92]. We give one example: Let $(N,+,\cdot)$ be a planar near-ring, take the elements of $N$ as points and take the set of $\{ a\cdot N^*+b\ \vert\ a,b\in N\}$ with $N^*=N\setminus\{0\}$ as blocks. Then the design $D$ with these points and blocks is a BIB-design.

   Some near-rings (Frobenius groups) yield circular designs, that is, designs where any two blocks intersect in at most two points. In this case, the corresponding near-rings (Frobenius groups) are called circular as well. The combinatorial structure of circular designs has been investigated in [Ke92,KK96]. As an example of the interplay between design and group structure, it is shown in [BFK96] that only Frobenius groups with metacyclic complement give circular designs. Moreover, in that paper it is shown that if $P\cdot\Phi$ is circular for a $p$-group $P$, then all Frobenius groups $G\cdot\Phi$ with $G$ a $p$-group (same $p$, same $\Phi$) are circular. The question remains open whether for any metacyclic group $\Phi$ which admits a fixed-point-free action, there exists a group $G$ such that $G\cdot\Phi$ yields a circular design.

   In [Pil96] applications of BIB-designs for statistical purposes (experimental design and analysis) are presented.

3. Codes from planar near-rings: Certain binary codes associated with (circular) designs coming from planar near-rings (Frobenius groups) reach the Johnson bound. These codes are optimal in the sense that they realize the maximal possible number of codewords with fixed weight, length and minimum distance [FHP90,Fuc91].

4. Structure theory of near-rings: In the theory of near-rings, fixed-point-free automorphism groups come into play at two different places.
   • All planar near-rings can be constructed via Ferrero's method [Fer70].
   • Every zero-symmetric $2$-primitive near-ring with unit is either a ring, or it is dense in a centralizer near-ring $M_{\Phi} (G)$, where $\Phi$ is a fixed-point-free automorphism group on $G$. The ideas used in the proof of this result have been adapted to $1$-primitive near-rings without unit in [FP89].
   Since every integral planar near-ring is $1$-primitive, it might be possible to see Ferrero's construction also coming from the characterization of $1$-primitive near-rings. Such considerations might yield:
   • Structural characterizations of planarity: For example, a finite zero-symmetric near-ring $N$ is integral and planar iff it is subdirectly irreducible and there is a natural number $n$ such that $x^n = x$ for all $x\in N$.
   • Geometrical structures: Planar near-rings give rise to geometrical structures, namely BIB-designs. It might be that similar structures can be obtained from other classes of $1$-primitive composition algebras.

5. Functions on Frobenius groups: In [Aic01] a formula for the number of polynomial functions on certain semi-direct products of groups is given. According to this result the number of polynomial functions $P(G\cdot\Phi)$ on a Frobenius group $G\cdot\Phi$ ($G$ normal) is equal to
   $$\vert P(G\cdot\Phi)\vert = \vert P(\Phi)\vert\cdot\vert R\vert^{\vert\Phi\vert}$$
   with $R=\{p\vert _G\ \vert\ p\in P(G\cdot\Phi)$ and $p(G)\subseteq G\}$. For a complete understanding of the polynomial functions of $G\cdot\Phi$, the set $R$ has to be investigated further. If the Frobenius kernel $G$ is abelian, then the functions in $R$ are of the form $x \mapsto s x + g$ ($g \in G$), where $s$ is an element of the group ring $\mathbb{Z}[\Phi]$.

6. Computing with near-rings: A computer algebra package for near-rings (``SONATA")[ABE$^+$00] based on GAP was developed in Linz (FWF-Projects P11486-TEC and P12911-INF). This allows us to study planar near-rings and designs on a computer. SONATA has already established itself as an important tool in near-ring theory.