In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.
Structures and operations edit
Basic properties of groups edit
- Butterfly lemma
- Center of a group
- Centralizer and normalizer
- Characteristic subgroup
- Commutator
- Composition series
- Conjugacy class
- Conjugate closure
- Conjugation of isometries in Euclidean space
- Core (group)
- Coset
- Derived group
- Euler's theorem
- Fitting subgroup
- Hamiltonian group
- Identity element
- Lagrange's theorem
- Multiplicative inverse
- Normal subgroup
- Perfect group
- p-core
- Schreier refinement theorem
- Subgroup
- Transversal (combinatorics)
- Torsion subgroup
- Zassenhaus lemma
Group homomorphisms edit
Basic types of groups edit
Simple groups and their classification edit
- Alternating group
- Borel subgroup
- Chevalley group
- Conway group
- Feit–Thompson theorem
- Fischer group
- General linear group
- Group of Lie type
- Group scheme
- HN group
- Janko group
- Lie group
- Linear algebraic group
- List of finite simple groups
- Mathieu group
- Monster group
- Projective group
- Reductive group
- Simple group
- Special linear group
- Symmetric group
- Thompson group (finite)
- Tits group
- Weyl group
Permutation and symmetry groups edit
- Arithmetic group
- Braid group
- Burnside's lemma
- Cayley's theorem
- Coxeter group
- Crystallographic group
- Crystallographic point group, Schoenflies notation
- Discrete group
- Euclidean group
- Even and odd permutations
- Frieze group
- Frobenius group
- Fuchsian group
- Geometric group theory
- Group action
- Homogeneous space
- Hyperbolic group
- Isometry group
- Orbit (group theory)
- Permutation
- Permutation group
- Rubik's Cube group
- Space group
- Stabilizer subgroup
- Steiner system
- Strong generating set
- Symmetry
- Symmetric group
- Symmetry group
- Wallpaper group
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Mathematical objects making use of a group operation edit
Mathematical fields and topics making important use of group theory edit
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Group representations edit
Computational group theory edit
Applications edit
Famous problems edit
Other topics edit
- Amenable group
- Capable group
- Commensurability (group theory)
- Compact group
- Compactly generated group
- Complete group
- Complex reflection group
- Congruence subgroup
- Continuous symmetry
- Frattini subgroup
- Growth rate
- Heisenberg group, discrete Heisenberg group
- Molecular symmetry
- Nielsen transformation
- Reflection group
- Tarski monster group
- Thompson groups
- Tietze transformation
- Transfer (group theory)
Group theorists edit
- N. Abel
- M. Aschbacher
- R. Baer
- R. Brauer
- W. Burnside
- R. Carter
- A. Cauchy
- A. Cayley
- J.H. Conway
- R. Dedekind
- L.E. Dickson
- M. Dunwoody
- W. Feit
- B. Fischer
- H. Fitting
- G. Frattini
- G. Frobenius
- E. Galois
- G. Glauberman
- D. Gorenstein
- R.L. Griess
- M. Hall, Jr.
- P. Hall
- G. Higman
- D. Hilbert
- O. Hölder
- B. Huppert
- K. Iwasawa
- Z. Janko
- C. Jordan
- F. Klein
- A. Kurosh
- J.L. Lagrange
- C. Leedham-Green
- F.W. Levi
- Sophus Lie
- W. Magnus
- E. Mathieu
- G.A. Miller
- B.H. Neumann
- H. Neumann
- J. Nielson
- Emmy Noether
- Ø. Ore
- O. Schreier
- I. Schur
- R. Steinberg
- M. Suzuki
- L. Sylow
- J. Thompson
- J. Tits
- Helmut Wielandt
- H. Zassenhaus
- M. Zorn