Group Theory 0 (Preface + Table of Contents)

Welcome. In this series of notes, I’ll be covering some basic materials in Group Theory. Ok, truth is, even though I said “basic”, there’ll probably be enough materials to cover two semesters of undergraduate algebra. The aim of this set of notes is to explain the various concepts in a more relaxed / informal setting, without sacrificing too much mathematical rigour. Needless to say, the approach is entirely subjective and the astute reader may soon learn to develop his own viewpoint.

This set of notes is not intended to completely replace a textbook. Many of the proofs are omitted or contracted, there’re not enough exercises, and it’s certainly not healthy to stick to a single point-of-view.

Table of Contents

Chapter I presents a first look at the best example of groups: the set of permutations, before plunging the reader into the abstract notion of groups.

Chapter II is a first look at abstract groups, with some basic concepts like cancellation laws, order of group/group elements. Examples are also provided for both finite & infinite groups.

Chapter III introduces the concept of subgroups: subsets of a group which inherits the group structure.

Chapter IV talks about cosets. Unlike subsets of a set, a subgroup actually partitions a group nicely as a disjoint union of equal-sized pieces, called cosets.

Chapter V talks about normal subgroups, where the cosets themselves actually form a group – called the quotient group.

Chapter VI introduces homomorphisms:  a function from one group to another which respects the underlying group operation.

Chapter VII follows an abstract group back to its “roots” by looking at its elements as permutations on a certain set. The key result is the Class Formula! We can’t stress this enough.

Chapter VIII presents the crown jewel of finite group theory: the three Sylow theorems. The results are exceptionally beautiful and useful!

Chapter IX discusses the long-standing pipedream of group theorists: that of classifying groups up to isomorphism.

Chapter X introduces groups which are defined by generators and relations. Burnside’s problem presents an interesting application of such groups.

Chapter XI talks about solvable and nilpotent groups. Solvable groups are basically those which can be “built” from abelian groups by extension.

Chapter XII takes things to a higher level of abstraction, by taking a first look at category theory. In fact, you can skip here straight after Chapter VI if you’ve the stomach for it.

And finally, the epilogue.

 

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