On Finite Circular Planar Nearrings Original Research Article

The main subject of this paper is to characterize finite circular Ferrero pairs, the tool for constructing finite circular planar nearrings. A general characterization of finite circular Ferrero pairs is given. In particular, a generalized version of Modisettʹs characterization (see J. R. Clay “Nearrings: Geneses and Applications,” pp. 68–75, Oxford Univ. Press, Oxford, 1992) is presented for finite circular Ferrero pairs (N, Φ) with cyclic Φ. We show that the fixed point free group of automorphisms Φ of a finite circular Ferrero pair (N, Φ) is metacyclic. Finally, two questions on the existence of nonabelian circular planar nearrings are answered by a general construction method and some basic examples.