On finite circular planar nearrings
Every planar nearring gives rise to a Ferrero pair. That is a pair \((N,\Phi)\) of groups such that \(\Phi\) acts on \(N\) as a group of automorphisms subject to the condition: \(-\phi+1\) is bijective if \(\phi\in\Phi\setminus\{1\}\). If \(N\) is finite, this just means that \(\Phi\) acts fixed point freely. \((N,\Phi)\) as well as the corresponding nearrings are called circular if \(\Phi a\cap\Phi b+c\) has at most two elements whenever \(a,b,c\in N\setminus\{0\}\). A first pioneering result was obtained by \textit{M. C. Modisett} [Util. Math. 35, 83-94 (1989; Zbl 0667.05007)]: if \(N\) is the additive group of a finite field and \(\Phi\) a subgroup of the multiplicative group, then circularity depends only on the characteristic \(p\) of that field. Moreover, the set \({\mathcal P}_\Phi\) of primes \(p\), for which \((N,\Phi)\) is not circular, is finite. Last, but not least, Modisett gives an algorithm to compute \({\mathcal P}_\Phi\). Note that \(\Phi\) is necessarily cyclic in the case considered by Modisett. This result was slightly generalized by Ke and the reviewer to arbitrary elementary abelian \(p\)-groups \(N\). The paper under review makes a decisive step ahead. Firstly, it is shown that for arbitrary finite \(p\)-groups, circularity again depends on \(p\) alone. So it makes sense to study the set \({\mathcal P}_\Phi\) of exceptional primes as before. This result is most general, since by a famous result of Thompson, \(N\) is nilpotent for every finite Ferrero pair \((N,\Phi)\) if \(\Phi\) is non-trivial. Therefore, \(N\) is a direct product of it's Sylow-subgroups. This reduces questions about circularity to \(p\)-groups. Secondly, the authors study the sets \({\mathcal P}_\Phi\) for various groups. If \(\Phi\) contains the quaternion group of order 8, then \({\mathcal P}_\Phi\) is the set of all primes, i.e., \((N,\Phi)\) is never circular. (The proof of theorem (3.1) shows just this, although the statement seems to be weaker.) As a consequence, if \((N,\Phi)\) is circular, then the Sylow-subgroups of \(\Phi\) are all cyclic, \(\Phi\) is metacyclic and can be described by generators and relations in a well-known way. If for a given group \(\Phi\) there exists a group \(N\), such that \((N,\Phi)\) is circular, then \({\mathcal P}_\Phi\) is a finite set. The authors show by examples that there do exist many such \(\Phi\)'s. It remains open, whether for any metacyclic group \(\Phi\) which admits a fixed point free action, there exists a group \(N\), such that \((N,\Phi)\) is circular. (Also submitted to MR).
scientific article; zbMATH DE number 954779
scientific article