List of group theory topics This is a list of group theory topics , by Wikipedia page , to provide an overview of the topic and to allow those interested to use the `` Related changes '' feature to keep abreast of edits in this area . See also : List of abstract algebra topics , List of category theory topics , list of Lie group topics . History , major contributors Joseph-Louis Lagrange Niels Abel Évariste 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 Horton Conway Famous problems Classification of finite simple groups Word problem for groups Subset sum problem Burnside 's problem Whitehead problem Herzog-Schönheim conjecture Applications Computer algebra system Exponentiating by squaring Knapsack problem Shor 's algorithm Cryptography Discrete logarithm Triple DES Caesar cipher Standard Model 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 Quaternion group Hilbert space Pauli matrices Gell-Mann matrices Banach-Tarski paradox Galois group Dimensional analysis Lie group Algebraic group Elliptic curve Abelian variety Group object Category of groups Abstract algebra Field Finite field Galois theory Group with operators Linear algebra Magma Module Monoid Monoid ring Quandle Quasigroup Ring Semigroup Vector space 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 Transversal Lagrange 's theorem Euler 's theorem Normal subgroup Hamiltonian group Characteristic subgroup Torsion subgroup Centralizer and normalizer Center of a group Commutator Derived group Conjugation of isometries in Euclidean space Conjugacy class Conjugate closure Core ( group ) Composition series Zassenhaus lemma , Butterfly lemma Schreier refinement theorem Fitting subgroup , generalized Fitting subgroup p-core Group homomorphisms Automorphism Automorphism group Group homomorphism Group isomorphism Factor group Quotient group Homomorphism Order automorphism Isomorphism theorem Fundamental theorem on homomorphisms Inner automorphism Outer automorphism group Basic group structures and operations Generating set of a group Sylow theorem Hall subgroup Product of group subsets Direct product Direct sum Direct sum of groups Subdirect sum Semidirect product Group extension Central extension Extension problem Schur multiplier Group cohomology Wreath product Free product Free group Free abelian group Presentation of a group Basic building block of groups , and classifications Examples of groups P-group Cyclic group Abelian group Rank of an abelian group Finitely generated abelian group Nilpotent group Solvable group Dihedral group Dicyclic group Klein four-group Group representation Pro-finite group Locally cyclic group List of small groups Divisible group Simple groups and their classification Simple group Quasisimple group Symmetric group Alternating group General linear group Special linear group Projective group Mathieu group Janko group Conway group Fischer group Thompson group ( finite ) Tits group Baby Monster group Monster group Bimonster Lie group Simple Lie group Weyl group Algebraic group Borel subgroup Parabolic subgroup Group scheme Group of Lie type , Chevalley group Feit-Thompson theorem Understanding groups as permutation and symmetry groups Rubik 's Cube Symmetry Symmetry group Euclidean group Frieze group Wallpaper group Crystallographic group Crystallographic point group , Schoenflies notation Space group Coxeter group Discrete group Frobenius group Fuchsian group Arithmetic group Homogeneous space Permutation Permutation group Symmetric group Cayley 's Theorem Even and odd permutations Strong generating set Braid group Group action Orbit ( group theory ) Stabilizer subgroup Burnside 's lemma Steiner system Geometric group theory Group representations See also list of harmonic analysis and representation theory topics Representation theory Affine representation Projective representation Maschke 's theorem Schur 's lemma Character theory Monstrous moonshine Computational group theory Coset enumeration Schreier-Sims algorithm Todd-Coxeter algorithm Schreier 's subgroup lemma Other topics Continous symmetry Frattini subgroup Transfer ( group theory ) Growth rate Heisenberg group , discrete Heisenberg group Congruence subgroup Commensurability ( mathematics ) Compactly generated group Complete group Thompson groups Tarski monster group Amenable group Compact group Covering a group by cosets or subgroups Categories : Mathematics-related lists | Abstract algebra | Group theory In other languages : Deutsch 