# An Introduction To Solvable, Supersolvable, And Nilpotent Groups

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## Authors

Curry, James

## Date of Issue

1974-04-01

## Type

thesis

## Language

## Subject Keywords

## Other Titles

## Abstract

Solvability and nilpotence form the core of this work. Appropriate to the undergraduate level, it begins with a review of the basic results of a first course in abstract algebra. These are extended to the Sylow Theorems, important because of their treatment of the existence of subgroups in a given group. Given this introduction, the solvability property is postulated in the form of the existence of p-complements within the given group. While this is not the conventionally used definition, it seems to have produced some simplification of the proofs of the standard theorems of solvability. Much more emphasis is given to nilpotence than to supersolvability. The relationship between the two properties is established, and most of the classical properties of nilpotence are given. This discussion is somewhat less exhaustive than the investigation of solvability. Both solvability and nilpotence find good application in the study of p-groups and in the Galois theory. Since at least one p-group occurs in any group (except the identity), their study is of obvious importance and provides a useful example of a solvable group. In studying the automorphisms of a given field, it becomes obvious that the rather minimal postulates of group theory can given startlingly important results when applied to much larger systems. It is, from a personal standpoint, very satisfying to have completed this paper. While it presents no really original research results, it has afforded a chance to investigate a very beautiful topic in mathematics in some depth. The writing of the paper has afforded a unique opportunity to
study the axiomatic structure of group theory. It is to he regretted (a little) that no original problem of any significance has been explored in these pages.
Few readers, if any at all, will be primarily interested in group theory. It is to be hoped that those few will find here an adequate basis to extend further undergraduate "research" toward the Galois theory or perhaps toward a more particularized section of group theory. I hope that the others find a topic as fascinating to them as group theory has been to me. Solvability and nilpotence form the core of this work. Appropriate to the undergraduate level, it begins with a review of the basic results of a first course in abstract algebra. These are extended to the Sylow Theorems, important because of their treatment of the existence of subgroups in a given group. Given this introduction, the solvability property is postulated in the form of the existence of p-complements within the given group. While this is not the conventionally used definition, it seems to have produced some simplification of the proofs of the standard theorems of solvability. Much more emphasis is given to nilpotence than to supersolvability. The relationship between the two properties is established, and most of the classical properties of nilpotence are given. This discussion is somewhat less exhaustive than the investigation of solvability. Both solvability and nilpotence find good application in the study of p-groups and in the Galois theory. Since at least one p-group occurs in any group (except the identity), their study is of obvious importance and provides a useful example of a solvable group. In studying the automorphisms of a given field, it becomes obvious that the rather minimal postulates of group theory can given startlingly important results when applied to much larger systems. It is, from a personal standpoint, very satisfying to have completed this paper. While it presents no really original research results, it has afforded a chance to investigate a very beautiful topic in mathematics in some depth. The writing of the paper has afforded a unique opportunity to
study the axiomatic structure of group theory. It is to he regretted (a little) that no original problem of any significance has been explored in these pages.
Few readers, if any at all, will be primarily interested in group theory. It is to be hoped that those few will find here an adequate basis to extend further undergraduate "research" toward the Galois theory or perhaps toward a more particularized section of group theory. I hope that the others find a topic as fascinating to them as group theory has been to me.