Raievska M. Multiplicative groups of finite local nearrings

In the thesis the multiplicative group of local nearrings are investigated. Structure theorems for nearfields with multiplicative group which is a Schmidt, Miller-Moreno or hereditary non-abelian group are obtained. A detailed structure of a local nearring whose subgroup of non-invertible elements is non-trivial cyclic is described. The local nearrings with multiplicative Miller-Moreno groups contained in the library of all nearrings with identity of the SONATA package of the system of computer algebra GAP are classyfied up to isomorphism. Based on GAP, an algorithm for finding all groups $K$ of order $2^n$ with an automorphism group $A$ of order $2^{n-1}$ having on $K$ an orbit of length $2^{n-1}$ whose complement in $K$ is a subgroup of $K$ is constructed. Using this algorithm, all non-isomorphic groups $K$ of order $32$ and $64$ with the latter property are found. A full classification of local nearrings of odd order with multiplicative Miller-Moreno group are given. The structure of local nearrings with multiplicative Miller-Moreno group which is not a $2$--group is described in detail. It is proved that a local nearring whose multiplicative group is a metacyclic $2$--group is of order at most $32$.