Encyclopedia > List of group theory topics
List of group theory topics
This order is by no means ideal; but it may help to provide some framework for a re-editing of the main topic. Nor are the entries below considered a definitive set of topics; rather they reflect the current state of group theory articles on Wikipedia. Feel free to add or subtract from this list as appropriate.
History of; major contributors to
- Joseph-Louis Lagrange
- Niels Abel
- Evariste Galois
- Augustin Louis Cauchy
- Arthur Cayley[?]
- Otto Ludwig Hölder[?]
- Camille Jordan
- Ludwig Sylow[?]
- Ferdinand Georg Frobenius[?]
- Sophus Lie
- Felix Klein
- William Burnside
- Richard Dedekind
- David Hilbert
- Max August Zorn
- John Thompson[?]
- Martin Dunwoody
- John Conway
Famous Problems in
- Classification of finite simple groups
- Word problem for groups
- Subset sum problem
- Burnside's problem
- Whitehead problem
Applications of
- Computer algebra system
- Exponentiating by squaring
- Knapsack problem
- Shor's algorithm
- Cryptography
- Discrete logarithm
- Triple DES
- Caesar cipher
Definitions in common with other mathematical ideas
- Binary operation
- Bilinear operator
- Commutative
- Associativity
- Equivalence relation
- Equivalence class
- Bijection
- Multiplication table
- Lattice
- Up to
- Congruence relation
- Prime number
Mathematical objects which have (or make use of) a group operation
- Number
- Real number
- Integer
- Modular arithmetic
- Matrix
- Tensor
- Quaternion
- Hilbert space
- Pauli matrices
- Banach-Tarski paradox
- Natural number
- Dimensional analysis
- Manifold
- Lie group
- Algebraic group
- Elliptic curve
- Abelian variety
Category theory and
- Morphism
- Mathematical topos
- Groupoid
- Limit (category theory)
- Preadditive category
- Initial object
- Yoneda lemma
- Natural transformation
- Embedding
- Standard Model
Abstract Algebra and
- Abstract algebra
- Universal algebra
- Algebra
- Subalgebra
- Quasigroup
- Semigroup
- Monoid
- Monoid ring
- Module
- Ring (algebra)
- Finite field
- Field
- Vector space
- Linear algebra
- Galois theory
- Galois group
Other mathematical disciplines which make great use of groups
- Geometry
- Algebraic geometry
- Algebraic topology
- Topological group
- Fundamental group
- Discrete space
- Homology
- Minkowski's theorem
Basic properties of groups and subgroups
- Elementary group theory
- Identity element
- Reciprocal
- Subgroup
- Coset
- Theorem of Lagrange
- Euler's theorem
- Normal subgroup
- Characteristic subgroup
- Torsion subgroup
- Centralizer and normalizer
- Center of a group
- Commutator
- Derived group
- Conjugate
- Conjugacy class
- Conjugate closure
- Core (group)[?]
Group Homomorphisms
- Automorphism
- Automorphism group
- Group homomorphism
- Group isomorphism
- Factor group
- Quotient group
- Homomorphism
- Order automorphism
- Isomorphism theorem
- Fundamental theorem on homomorphisms
- Inner automorphism
Basic group structures and operations
- Generating set of a group
- Sylow theorem
- Product of groups
- Direct product
- Direct sum
- Direct sum of groups
- Subdirect sum[?]
- Semidirect product
- Group Extension
- Group cohomology
- Wreath product
- Free product[?]
- Free group
- Free abelian group
- Presentation of a group
Basic Building Block of groups and classifications
- Simple group
- P-group
- Cyclic group
- Abelian group
- Nilpotent group
- Solvable group
- Dihedral group
- Klein four-group
- Symmetric group
- Alternating group
- General linear group
- Special linear group
- Group representation
- Pro-finite group
- Monster group
- Lie group
- List of small groups
- Finitely generated abelian group
- Divisible group[?]
Understanding groups as permutation groups
- Rubik's Cube
- Symmetry
- Symmetry group
- Permutation
- Even permutation
- Permutation group
- Group action
- Cayley's Theorem
- Steiner system
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