Group theory - Table of contents

This is a table of contents for my notes on Group theory. The posts are arranged in an order that might be found in a textbook.

Elements of set theory

• Identities in set theory
• Mathematical induction
• Mathematical induction with inequality
• Binary relations on sets
• Equivalence and order relations on sets
• Equivalence classes on sets
• Functions as relations on sets
• Injective, surjective and bijective functions

Algebraic operations on integers

• Quotient-remainder theorem
• Quotient-remainder theorem examples
• Euclidean algorithm for greatest common divisor
• Bézout's lemma for the greatest common divisor
• Bézout's lemma - forward algorithm
• Linear Diophantine equations
• Common divisors of linear combination of integers
• Least common multiple and greatest common divisor
• Relatively prime integers
• Prime numbers
• Fundamental theorem of arithmetic

The integers modulo n

• Integers modulo n
• Divisibility using modulo n
• Modulo n as an equivalence relation
• Functions of modulo n
• Addition and multiplication modulo n
• Linear equations with modulo n

Semigroups and monoids

• Binary operations on sets
• Identity elements in binary operations
• Inverses in binary operations
• Semigroups and monoids
• Idempotent elements in semigroups

Groups

• Groups - definition and examples
• Cayley tables for finite groups

Definition and properties of groups

• Definition of a group
• Subgroups and Lagrange's theorem
• Group multiplication tables and order-6 groups
• Direct products of groups
• Permutation groups - cycles and transpositions
• Cayley's theorem
• Equivalence classes
• Cycle structure of permutation groups and partition of integers
• Equivalence (conjugacy) classes of permutation groups
• Dihedral groups
• Equivalence (conjugacy) classes of dihedral groups
• Quaternions as groups
• Quaternion arithmetic
• Vectors as quaternions
• Exponential of a quaternion
• Rotations using quaternions
• Rotations in general using quaternions
• Coxeter groups
• Invariant subgroups
• Derived subgroups
• Cosets and quotient groups
• Sets, groups, fields, vector spaces and algebras

Continuous groups

• Lie groups, Lie algebras and rotations
• Lie algebra of the rotation group SO(n)
• Representation of rotation group with rank-2 tensors
• Irreducible representations of rotation group
• Invariant symbols and dual tensors

Representation theory

• Representations and characters of groups
• Regular representations of groups
• Unitarity theorem for groups
• Two-dimensional representations of the permutation group
• Schur's lemma
• Great orthogonality theorem
• Character orthogonality
• Sizes of irreducible representations
• Class algebra
• Character table for S₄
• Character table for S₄ using the regular representation
• Real, pseudoreal and complex representations
• Test for real, pseudoreal and complex representations
• Square roots of group elements

Number theory

• Lemma for relatively prime numbers
• Group of relatively prime numbers