Modern Groups & Representations

Contents

Highlights

• Basics
• Groups
• Representations

This wiki book is based on Wikipedia articles. No originality is claimed. The purpose of this book is to provide a pedagogical arrangement of the topics in Group theory and representation theory. This means it starts from the very basics to the advanced (and contemporary) topics in these subjects. The primary readers would be mathematical (or theoretical) physicists and mathematicians. But, any interested readers may get benefitted from this book.

Basics

• Mathematical object

• Number System

  • Set
    • Element
    • Binary Relation
      • Reflexive Relation
      • Symmetric Relation
      • Transitive Relation
      • Equivalence Relation
      • Partial Order $\mid$ Partially Ordered Set (poset)
      • Total Order
    • Map
      • Domain
      • Co-domain
      • Range
      • Image
      • Injective $\mid$ one-to-one $\mid$ 1-1
      • Surjective $\mid$ onto
      • Bijective $\mid$ one-to-one and onto $\mid$ one-to-one correspondence $\mid$ 1-1 correspondence $\mid$ invertible
      • Homomorphism
        • Kernel
  • Operand
  • Binary Operation
    • Commutative
    • Associative
    • Distributive
  • Algebraic structure
  • Point Set Topology
    • Open Set
    • Closed Set
    • Bounded Set
    • Compactness
    • Topological space
      • Connected

• Field

  • Finite Field $\mid$ Galois field
    • Factorization of polynomials over finite fields
  • Field extension
    • Algebraic extension
  • Algebra
    • Unital algebra
    • Zero algebra
    • Associative algebra
    • Distributive algebra $\mid$ Non-associative algebra

• Ring

  • Integral Domain
    • Gaussian integer
  • Quotient ring $\mid$ Factor Ring
  • Fermat’s Little Theorem
  • Euler’s Theorem
  • Ideal
    • Prime Ideal
    • Maximal Ideal
    • Principal ideal
      • Principal ideal domain
    • Unique factorization domain $\mid$ Factorial Ring
    • Euclidean domain $\mid$ Euclidean ring

• Vector space

  • Basis
  • Dual basis
  • Dimension
  • Linear combination
    • Linear independence
    • Span
  • Bilinear map
  • Sesquilinear form $\mid$ Hermitian Product
  • Inner product $\mid$ Scalar product
  • Tensor product
  • Subspace
    • Quotient Space
  • Lie Algebra

• Module

Groups

• Semigroup

• Monoid

• Group

  • Abelian Group
  • Generator
  • Finite Group
    • Order of a Group
    • Trivial Group
    • Cayley table
    • Cyclic Group
    • Permutation Group
      • Alternating group
    • Symmetric Group
  • Infinite (Discrete) Group
    • Modular Group
  • (Continuous) Compact Group
    • Orthogonal Group $O(n)$
      • Special Orthogonal Group $SO(n)$
    • Unitary Group $U(n)$
      • Special Unitary Group $SU(n)$

• Sub-group

  • coset
  • Conjugates $\mid$ Conjugacy Class
  • Center
  • Normal Subgroup
    • Simple Group
    • Quotient Group
    • Lagrange theorem
  • Centralizer and normalizer

• Group Homomorphism $\mid$ Not to be confused with homeomorphism

  • Group Monomorphism $\mid$ Injective Homomorphism
  • Group Epimorphism $\mid$ Surjective Homomorphism
  • Group Isomorphism $\mid$ Bijective Homomorphism
    • Cayley’s Theorem
    • Isomorphism Theorem
      • First Isomorphism Theorem $\mid$ Fundamental Theorem of Homomorphism $\mid$ See its proof at page 18 of Lüdeling’s note.
  • Group Endomorphism $\mid$ Homomorphism from a group to itself
  • Group Automorphism $\mid$ Bijective Endomorphism
    • Inner automorphism
    • Outer automorphism group

• Product of Groups

  • Direct Product
    • Kronecker product $\mid$ matrix direct product
  • Direct Sum
  • Free Product
  • Semidirect Product

Representations

• Group Action