Date of Award

5-1970

Degree Type

Master's Essay - Restricted

Degree Name

Master of Science (MS)

Department

Mathematical and Statistical Sciences

First Advisor

James Sauve

Abstract

The concept of a topological group and in particular that of a Lie group arose historically in the study of transformation groups acting on a Euclidean space. But later it was discovered that the concepts are much broader and can be studied in their own right without referring back to transformations. From this point of view the study of a topological group combines concepts of abstract group theory and of topology.

The purpose of this paper is to discuss some elementary properties of Lie groups. The main object of the study will be the local Lie group rather than the global structure. Assumed is a basic understanding of abstract algebra and topology.

The paper is divided into two parts; part I is an introduction to general topological groups, with emphasis on those results that are of importance to the later discussion. The definitions follow Pontryagin and most of the theorems are stated without proof. Part II contains the main development: the discussion of the local Lie group. It has four sections. The first defines the concept of a Lie group and discusses some elementary properties. Section two deals with the invariance theorem and its consequences. In section three two theorems are proved: that the subgroup and the factor group of a local Lie group are themselves local Lie groups. In the final section analytic manifolds are introduced and their relation to Lie groups is shown.

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