Finite Nilpotent Rings And The ρ-Groups Determined By Them

This thesis consists of three chapters. Chapter I and chapter II contain a summary of fundamental knowledge about finite p-groups. Chapte III is devoted to a discussion of the p-group determined by finite regular p-nilpotent rings.In section III.1, the concept of finite regular p-nilpotent rings is definied. It is shown that every finite p-nilpotent ring is the homomorphic image of a finite regular p-nilpotent ring, and every finite p-nilpotent ring R corresponds to a finite p-group-R*in a natural way. It is also shown that every ideal in a finite nilpotent ring R corresponds to a normal subgruops of R*. By the means of the linearity of the groups derived from finite regular p-nilpotent rings, some properties of such p-groups can be obtained. Theorem III.2.6gives an expression of such p-group by generators and relations. In theorem III.2.7and2.8, some important subgroups and subgroups series are characterized by the linear structure of rings. Section III.3gives a few examples of p-abel groups and mete-able groups. Section4～5discuss two special kinds of finite regular p-nilpotent rings and the groups derived from them, and some interesting properties of these p-groups are obtain. In section6some concrete finite regular p-nilpotent ring are constructed, it is shown that the groups derived from them are all indecomposable and are not isomorphic to each other; many finite regular p-nilpotent rings can be obtained by changing the combinatorial structure of these rings a little, it can be expected that one can obtain a lot of indecomposable p-groups from these rings and their ideals. In section III.7two Zp-represetation of the p-groups determined by finite p-nilpotent rings are set up, which makes it possible to study such p-groups by matrix methods, in proposition III.7.8, it is shown that every finite p-group is a subgroups of U(n, Zp) for some integer n. In section8, we try to generalize the concept of finite regular nilpotent rings. Doing so is just putting as many p-groups as possible under linear structure since the ideals in a finite regular p-nilpotent ring can be extremly complicated. In section Ⅲ.9, some p-groups of order p5and p6are constructed by finite regular p-nilpotent rings and their ideals. By using the generators and relations formulated in Theorem III.2.6. Some finite p-groups are constructed in section10. In the last section of chapter III, an additional topic, the sylow p-subgroups of GL(n, Zpr), is discussed. It is verified that the sylow p-groups of GL(n,Zpr) is regular if (?)<p.