| In this dissertation, substructures of endomorphism near-rings and rings of a group are considered, with emphasis on the connection between the substructure and the underlying group. In particular, homomorphisms from a group to a fixed subgroup are used to additively generate a near-ring structure. This near-ring is a “finer” tool than the endomorphism near-ring, for studying properties of the group. If the subgroup is proper, this near-ring does not contain the unity element of the endomorphism near-ring and may not have a unity element at all. The present work addresses the problem of characterizing, in terms of the group and its fixed proper subgroup, when this near-ring has a one-sided or two-sided unity element, which completely solves two problems posed in a 1997 paper by G. F. Birkenmeier, H. E. Heatherly, and G. Pilz. Also considered is when these near-rings generated by group homomorphisms into a fixed subgroup are rings, which is certainly a counter-intuitive possibility when the fixed subgroup is nonabelian.; This complements another area of research which focuses on another substructure of the endomorphism near-ring: the subset of distributive elements. Given that the set of endomorphisms of a group is contained in the set of distributive elements of its endomorphism near-ring which, in turn, is contained in the endomorphism near-ring, it is shown that the class of all groups is partitioned into four nonempty subclasses when all combinations of these inclusions, proper or nonproper, are considered. Some of these classes are well-known classes of groups. Furthermore, a characterization of each subclass is possible, in terms of the orbits of the underlying group.; A direct result of this research is the determination of when the endomorphism near-ring of a proper subgroup of a group embeds in the endomorphism near-ring of that group. Another consequence is the simplification of the well-known problem of determining when the semidirect product of two I-E groups is an I-E group. Finally, applications to the general study of E-groups are discussed. |
