Raievska I. The additive groups of finite local nearrings

It is well known that the main structures connected with a nearring $R$ with an identity are its additive group $R^+$, the multiplicative group $R^*$ and the set $L$ of all non-invertible elements of $R$. The nearring $R$ is called local if $L$ is a subgroup of $R^+$. The thesis is devoted to the study of finite local nearrings with Miller-Moreno additive groups, i.e. non-abelian groups whose all proper subgroups are abelian.In particular, relations between the structure of these local nearrings and that of their additive groups are investigated. Furthermore, conditions for Miller-Moreno groups to be the additive group of a finite nearring with identity are also found. The main result of the thesis is theorems giving the necessary and sufficient conditions of existence of finite local nearrings on the Miller-Moreno groups and the classification of these groups. The subgroups of all non-invertible elements of such local nearrings are characterized. Some formulas determining multiplication in the finite local nearrings with additive Miller-Moreno groups are given. Finally, algorithms for constructing such nearrings are developed. These algorithms are realized on Miller-Moreno groups of orders 32 and 64 by means of computer algebra system GAP and package SONATA.