Finite local nearrings

Abstract

Nearrings arise naturally in the study of systems of nonlinear mappings, and they have been studied for many decades. Basic definitions and many results concerning nearrings can be, for instance, found in [G. Pilz. Near-rings. The theory and its applications. North Holland, Amsterdam, 1977].Nearrings are generalized rings in the sense that the addition need not be commutative and only one distributive law is assumed. Clearly, every associative ring is a nearring, and each group is an additive group of a nearring, but not necessarily of a nearring with identity. The question what group can be an additive group of a nearring with identity is far from solution.A nearring with identity is called local if the set of all its non-invertible elements is a subgroup of its additive group. A study of local nearrings was initiated by Maxson (1968) who defined a number of their basic properties and proved, in particular, that the additive group of a finite zero-symmetric local nearring is a p-group. The determination of the non-abelian finite p-groups which are the additive groups of local nearrings is an open problem (Feigelstock, 2006).The list of all local nearrings of order at most 31 can be extracted from the package SONATA (https://www.gap-system.org/Packages/sonata.html) of the computer system algebra GAP (https:// www.gap-system.org/).We observe also that there exist 14 non-isomorphic groups of order 16 = 24 from which 9 are the additive groups of local nearrings. Groups of order 32 = 25 with this property are described. In particular, among 51 non-isomorphic groups of this order only 19 are these additive groups.In this paper finite local nearrings are studied. Moreover, local nearrings of order at most 32 are classified.