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.
- Group Theory I.1 (Introduction to Permutations)
- Group Theory I.2 (Notations for Permutations)
- Group Theory I.3 (Order of a Permutation)
- Group Theory I.4 (Conjugate Permutations)
- Group Theory I.5 (Parity of a Permutation)
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.
- Group Theory II.1 (The Axioms of a Group)
- Group Theory II.2 (Basic Concepts)
- Group Theory II.3 (Examples of Finite Groups)
- Group Theory II.4 (Examples of Infinite Groups)
Chapter III introduces the concept of subgroups: subsets of a group which inherits the group structure.
- Group Theory III.1 (Subgroups: Definition)
- Group Theory III.2 (Examples of Subgroups)
- Group Theory III.3 (Subgroups of Cyclic Groups)
- Group Theory III.4 (Properties of Subgroups)
- Group Theory III.5 (Generating Subgroup From Subset)
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.
- Group Theory IV.1 (Cosets)
- Group Theory IV.2 (Lagrange’s Theorem)
- Group Theory IV.3 (Examples of Lagrange’s Theorem)
- Group Theory IV.4 (Optional: Double Cosets)
Chapter V talks about normal subgroups, where the cosets themselves actually form a group – called the quotient group.
- Group Theory V.1 (Normal Subgroups)
- Group Theory V.2 (First Properties)
- Group Theory V.3 (Examples of Normal Subgroups)
- Group Theory V.4 (Optional: Automorphism Group)
- Group Theory V.5 (Products of Subgroups)
Chapter VI introduces homomorphisms: a function from one group to another which respects the underlying group operation.
- Group Theory VI.1 (Homomorphisms)
- Group Theory VI.2 (First Isomorphism Theorem)
- Group Theory VI.3 (Examples of Homomorphisms)
- Group Theory VI.4 (Second & Third Isomorphism Theorems)
- Group Theory VI.5 (Correspondence Between G and G/N)
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.
- Group Theory VII.1 (Group Actions)
- Group Theory VII.2 (Example of Group Action)
- Group Theory VII.3 (Class Formula)
- Group Theory VII.4 (Two More Examples)
Chapter VIII presents the crown jewel of finite group theory: the three Sylow theorems. The results are exceptionally beautiful and useful!
- Group Theory VIII.1 (Cauchy’s Theorem)
- Group Theory VIII.2 (First Sylow Theorem)
- Group Theory VIII.3 (Second Sylow’s Theorem)
- Group Theory VIII.4 (Third Sylow Theorem)
- Group Theory VIII.5 (Applications of Sylow Theorems)
Chapter IX discusses the long-standing pipedream of group theorists: that of classifying groups up to isomorphism.
- Group Theory IX.1 (Classification of Small Groups)
- Group Theory IX.2 (Finite Simple Groups)
- Group Theory IX.3 (The Semidirect Product)
- Group Theory IX.4 (Examples of Semidirect Products)
- Group Theory IX.5 (Classification of Abelian Groups)
Chapter X introduces groups which are defined by generators and relations. Burnside’s problem presents an interesting application of such groups.
- Group Theory X.1 (Free Groups)
- Group Theory X.2 (Presentation of a Group)
- Group Theory X.3 (Word Problem for Groups)
- Group Theory X.4 (Optional: Burnside’s Problem)
Chapter XI talks about solvable and nilpotent groups. Solvable groups are basically those which can be “built” from abelian groups by extension.
- Group Theory XI.1 (Solvable Groups)
- Group Theory XI.2 (Solvable Groups: Examples etc)
- Group Theory XI.3 (Nilpotent Groups)
- Group Theory XI.4 (Nilpotent Groups: Examples etc)
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.
- Group Theory XII.1 (Introduction to Diagrams)
- Group Theory XII.2 (Category Theory)
- Group Theory XII.3 (More Universal Properties)
- Group Theory XII.4 (Category Theory: Functors)
- Group Theory XII.5 (More Category Theory)
And finally, the epilogue.