Structures and operations

• Central extension
• Direct product of groups
  Direct product of groups
  In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
  
  
• Direct sum of groups
• Extension problem
• Free abelian group
  Free abelian group
  In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
  
  
• Free group
  Free group
  In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
  
  
• Free product
  Free product
  In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
  
  

• Generating set of a group
  Generating set of a group
  In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...
  
  
• Group cohomology
  Group cohomology
  In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
  
  
• Group extension
  Group extension
  In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...
  
  
• Presentation of a group
  Presentation of a group
  In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
  
  
• Product of group subsets

• Schur multiplier
  Schur multiplier
  In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
  
  
• Semidirect product
  Semidirect product
  In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
  
  
• Sylow theorems
  • Hall subgroup
    Hall subgroup
    In mathematics, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They are named after the group theorist Philip Hall.- Definitions :A Hall divisor of an integer n is a divisor d of n such that...
    
    
• Wreath product
  Wreath product
  In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. Wreath products are an important tool in the classification of permutation groups and also provide a way of constructing interesting examples of groups.Given two groups A and H...
  
  

Basic properties of groups

• Butterfly lemma
• Center of a group
• Centralizer and normalizer
  Centralizer and normalizer
  In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively...
  
  
• Characteristic subgroup
  Characteristic subgroup
  In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group. Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal...
  
  
• Commutator
  Commutator
  In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
  
  
• Composition series
  Composition series
  In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence...
  
  
• Conjugacy class
  Conjugacy class
  In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
  
  
• Conjugate closure
  Conjugate closure
  In group theory, the conjugate closure of a subset S of a group G is the subgroup of G generated by SG, i.e. the closure of SG under the group operation, where SG is the conjugates of the elements of S:The conjugate closure of S is denoted or G.The conjugate closure of any subset S of a group G...
  
  
• Conjugation of isometries in Euclidean space
  Conjugation of isometries in Euclidean space
  In a group, the conjugate by g of h is ghg−1.-Translation:If h is a translation, then its conjugate by an isometry can be described as applying the isometry to the translation:...
  
  

• Core (group)
  Core (group)
  In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p-core of a group.-Definition:...
  
  
• Coset
  Coset
  In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
  
  
• Derived group
• Elementary group theory
• Euler's theorem
• Fitting subgroup
  Fitting subgroup
  In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable...
  
  
  • Generalized Fitting subgroup
• Hamiltonian group
  Hamiltonian group
  In group theory, a Dedekind group is a group G such that every subgroup of G is normal.All abelian groups are Dedekind groups.A non-abelian Dedekind group is called a Hamiltonian group....
  
  
• Identity element
  Identity element
  In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
  
  

• Lagrange's theorem
  Lagrange's theorem (group theory)
  Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange....
  
  
• Multiplicative inverse
  Multiplicative inverse
  In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
  
  
• Normal subgroup
  Normal subgroup
  In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
  
  
• perfect group
  Perfect group
  In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients...
  
  
• p-core
• Schreier refinement theorem
  Schreier refinement theorem
  In mathematics, the Schreier refinement theorem of group theory states that any two normal series of subgroups of a given group have equivalent refinements....
  
  
• Subgroup
  Subgroup
  In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
  
  
• Transversal
  Transversal
  In geometry , when two coplanar lines exist such that a third coplanar line passes thru both lines. This third line is named the Transversal....
  
  
• Torsion subgroup
  Torsion subgroup
  In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order...
  
  
• Zassenhaus lemma
  Zassenhaus lemma
  In mathematics, the butterfly lemma or Zassenhaus lemma, named after Hans Julius Zassenhaus, is a technical result on the lattice of subgroups of a group or the lattice of submodules of a module, or more generally for any modular lattice....
  
  

Group homomorphisms

• Automorphism
  Automorphism
  In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
  
  
• Automorphism group
• Factor group
• Fundamental theorem on homomorphisms
  Fundamental theorem on homomorphisms
  In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism....
  
  
• Group homomorphism
  Group homomorphism
  In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
  
  
• Group isomorphism
  Group isomorphism
  In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
  
  

• Homomorphism
  Homomorphism
  In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
  
  
• Isomorphism theorem
  Isomorphism theorem
  In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures...
  
  
• Inner automorphism
  Inner automorphism
  In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
  
  
• Order automorphism
• Outer automorphism group
  Outer automorphism group
  In mathematics, the outer automorphism group of a group Gis the quotient Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out...
  
  
• Quotient group
  Quotient group
  In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
  
  

Basic types of groups

• Examples of groups
  Examples of groups
  Some elementary examples of groups in mathematics are given on Group .Further examples are listed here.-Permutations of a set of three elements:Consider three colored blocks , initially placed in the order RGB...
  
  
• Abelian group
  Abelian group
  In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
  
  
• Cyclic group
  Cyclic group
  In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
  
  
  • Rank of an abelian group
    Rank of an abelian group
    In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
    
    
• Dicyclic group
• Dihedral group
  Dihedral group
  In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
  
  
• Divisible group
  Divisible group
  In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n...
  
  
• Finitely generated abelian group
  Finitely generated abelian group
  In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the formwith integers n1,...,ns...
  
  

• Group representation
  Group representation
  In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
  
  
• Klein four-group
  Klein four-group
  In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...
  
  
• List of small groups
• Locally cyclic group
  Locally cyclic group
  In group theory, a locally cyclic group is a group in which every finitely generated subgroup is cyclic.-Some facts:*Every cyclic group is locally cyclic, and every locally cyclic group is abelian....
  
  
• Nilpotent group
  Nilpotent group
  In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
  
  
• Solvable group
  Solvable group
  In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
  
  
• P-group
  P-group
  In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
  
  
• Pro-finite group
  Pro-finite group
  In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups; they share many properties with their finite quotients.- Definition :...
  
  

Simple groups and their classification

• Classification of finite simple groups
  Classification of finite simple groups
  In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
  
  

• Algebraic group
  Algebraic group
  In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
  
  
• Alternating group
• Borel subgroup
  Borel subgroup
  In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the group GLn ,...
  
  
• Chevalley group
• Conway group
  Conway group
  In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway.The largest of the Conway groups, Co1, of order...
  
  
• Feit–Thompson theorem
  Feit–Thompson theorem
  In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by - History : conjectured that every nonabelian finite simple group has even order...
  
  
• Fischer group
  Fischer group
  In mathematics, the Fischer groups are the three sporadic simple groups Fi22, Fi23,Fi24' introduced by .- 3-transposition groups :...
  
  
• General linear group
  General linear group
  In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
  
  
• Group of Lie type
  Group of Lie type
  In mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
  
  

• Group scheme
  Group scheme
  In mathematics, a group scheme is a type of algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not...
  
  
• Janko group
  Janko group
  In mathematics, a Janko group is one of the four sporadic simple groups named for Zvonimir Janko. Janko constructed the first Janko group J1 in 1965. At the same time, Janko also predicted the existence of J2 and J3. In 1976, he suggested the existence of J4...
  
  
• Lie group
  Lie group
  In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
  
  
  • Simple Lie group
    Simple Lie group
    In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
    
    
• List of finite simple groups
• Mathieu group
  Mathieu group
  In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...
  
  
• Monster group
  Monster group
  In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
  
  
  • Baby Monster group
    Baby Monster group
    In the mathematical field of group theory, the Baby Monster group B is a group of orderThe Baby Monster group is one of the sporadic groups, and has the second highest order of these, with the highest order being that of the Monster group...
    
    
  • Bimonster
• Parabolic subgroup

• Projective group
• Simple group
  Simple group
  In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
  
  
  • Quasisimple group
    Quasisimple group
    In mathematics, a quasisimple group is a group that is a perfect central extension E of a simple group S...
    
    
• Special linear group
  Special linear group
  In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
  
  
• Symmetric group
  Symmetric group
  In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
  
  
• Thompson group (finite)
  Thompson group (finite)
  In the mathematical field of group theory, the Thompson group Th, found by and constructed by , is a sporadic simple group of orderThe centralizer of an element of order 3 of type 3C in the Monster group is a product of the Thompson group and a group of order 3, as a result of which the Thompson...
  
  
• Tits group
  Tits group
  In mathematics, the Tits group 2F4′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by ....
  
  
• Weyl group
  Weyl group
  In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
  
  

Permutation and symmetry groups

• Arithmetic group
  Arithmetic group
  In mathematics, an arithmetic group in a linear algebraic group G defined over a number field K is a subgroup Γ of G that is commensurable with G, where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them...
  
  
• Braid group
  Braid group
  In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...
  
  
• Burnside's lemma
  Burnside's lemma
  Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms include William Burnside, George...
  
  
• Cayley's Theorem
  Cayley's theorem
  In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...
  
  
• Coxeter group
  Coxeter group
  In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
  
  
• Crystallographic group
• Crystallographic point group
  Crystallographic point group
  In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...
  
  , Schoenflies notation
  Schoenflies notation
  The Schoenflies notation or Schönflies notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe Point groups. This notation is used in spectroscopy. The other convention is the Hermann–Mauguin notation, also known as the...
  
  
• Discrete group
  Discrete group
  In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
  
  
• Euclidean group
  Euclidean group
  In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...
  
  
• Even and odd permutations
  Even and odd permutations
  In mathematics, when X is a finite set of at least two elements, the permutations of X fall into two classes of equal size: the even permutations and the odd permutations...
  
  

• Frieze group
  Frieze group
  A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
  
  
• Frobenius group
  Frobenius group
  In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point.They are named after F. G. Frobenius.- Structure :...
  
  
• Fuchsian group
  Fuchsian group
  In mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...
  
  
• Geometric group theory
  Geometric group theory
  Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act .Another important...
  
  
• Group action
  Group action
  In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
  
  
• Homogeneous space
  Homogeneous space
  In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
  
  
• Isometry group
  Isometry group
  In mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...
  
  
• Orbit (group theory)
• Permutation
  Permutation
  In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
  
  
• Permutation group
  Permutation group
  In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
  
  

• Rubik's Cube
  Rubik's Cube
  Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik.Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 and won the German Game of the Year special award for Best Puzzle that...
  
  
• Space group
  Space group
  In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
  
  
• Stabilizer subgroup
• Steiner system
  Steiner system
  250px|right|thumbnail|The [[Fano plane]] is an S Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line....
  
  
• Strong generating set
  Strong generating set
  In abstract algebra, especially in the area of group theory, a strong generating set of a permutation group is a generating set that clearly exhibits the permutation structure as described by a stabilizer chain...
  
  
• Symmetry
  Symmetry
  Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
  
  
• Symmetric group
  Symmetric group
  In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
  
  
• Symmetry group
  Symmetry group
  The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
  
  
• Wallpaper group
  Wallpaper group
  A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
  
  

Concepts groups share with other mathematics

• Associativity
  Associativity
  In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
  
  
• Bijection
  Bijection
  A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
  
  
• Bilinear operator
  Bilinear operator
  In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments. Matrix multiplication is an example.-Definition:...
  
  
• Binary operation
  Binary operation
  In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
  
  
• Commutative
• Congruence relation
  Congruence relation
  In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...
  
  
• Equivalence class
• Equivalence relation
  Equivalence relation
  In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
  
  
• Lattice (group)
  Lattice (group)
  In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
  
  
• Lattice (discrete subgroup)
  Lattice (discrete subgroup)
  In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
  
  
• Multiplication table
  Multiplication table
  In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system....
  
  
• Prime number
  Prime number
  A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
  
  
• Up to
  Up to
  In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
  
  

Mathematical objects making use of a group operation

• Abelian variety
  Abelian variety
  In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
  
  
• Algebraic group
  Algebraic group
  In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
  
  
• Banach-Tarski paradox
• Category of groups
  Category of groups
  In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
  
  
• Dimensional analysis
  Dimensional analysis
  In physics and all science, dimensional analysis is a tool to find or check relations among physical quantities by using their dimensions. The dimension of a physical quantity is the combination of the basic physical dimensions which describe it; for example, speed has the dimension length per...
  
  
• Elliptic curve
  Elliptic curve
  In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
  
  
• Galois group
  Galois group
  In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
  
  

• Gell-Mann matrices
  Gell-Mann matrices
  The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the infinitesimal generators of the special unitary group called SU....
  
  
• Group object
  Group object
  In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets...
  
  
• Hilbert space
  Hilbert space
  The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
  
  
• Integer
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
  
• Lie group
  Lie group
  In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
  
  
• Matrix
• Modular arithmetic
  Modular arithmetic
  In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
  
  

• Number
  Number
  A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
  
  
• Pauli matrices
  Pauli matrices
  The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
  
  
• Real number
  Real number
  In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
  
  
• Quaternion
  Quaternion
  In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
  
  
  • Quaternion group
    Quaternion group
    In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
    
    
• Tensor
  Tensor
  Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
  
  

Mathematical fields & topics making important use of group theory

• Algebraic geometry
  Algebraic geometry
  Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
  
  
• Algebraic topology
  Algebraic topology
  Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
  
  
• Discrete space
  Discrete space
  In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
  
  
• Fundamental group
  Fundamental group
  In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
  
  
• Geometry
  Geometry
  Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
  
  
• Homology
  Homology (mathematics)
  In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
  
  
• Minkowski's theorem
  Minkowski's theorem
  In mathematics, Minkowski's theorem is the statement that any convex set in Rn which is symmetric with respect to the origin and with volume greater than 2n d contains a non-zero lattice point...
  
  
• Topological group
  Topological group
  In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
  
  

Algebraic structures related to groups

• Field
  Field (mathematics)
  In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
  
  
• Finite field
  Finite field
  In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
  
  
• Galois theory
  Galois theory
  In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
  
  
• Group ring
  Group ring
  In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
  
  
• Group with operators
  Group with operators
  In abstract algebra, a branch of pure mathematics, the algebraic structure group with operators or Ω-group is a group with a set of group endomorphisms.Groups with operators were extensively studied by Emmy Noether and her school in the 1920s...
  
  
• Heap
  Heap (mathematics)
  In abstract algebra, a heap is a mathematical generalisation of a group. Informally speaking, a heap is obtained from a group by "forgetting" which element is the unit, in the same way that an affine space can be viewed as a vector space in which the 0 element has been "forgotten"...
  
  
• Linear algebra
  Linear algebra
  Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
  
  
• Magma
  Magma (algebra)
  In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
  
  

• Module
  Module (mathematics)
  In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
  
  
• Monoid
  Monoid
  In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
  
  
• Monoid ring
• Quandle
• Quasigroup
  Quasigroup
  In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible...
  
  
• Ring
  Ring (mathematics)
  In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
  
  
• Semigroup
  Semigroup
  In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
  
  
• Vector space
  Vector space
  A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
  
  

Group representations

• Affine representation
  Affine representation
  An affine representation of a topological group G on an affine space A is a continuous group homomorphism from G to the automorphism group of A, the affine group Aff...
  
  
• Character theory
  Character theory
  In mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
  
  
• Great orthogonality theorem
• Maschke's theorem
  Maschke's theorem
  In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces...
  
  
• Monstrous moonshine
  Monstrous moonshine
  In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...
  
  
• Projective representation
  Projective representation
  In the mathematical field of representation theory, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear groupwhere GL is the general linear group of invertible linear transformations of V over F and F* here is the...
  
  
• Representation theory
  Representation theory
  Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
  
  
• Schur's lemma
  Schur's lemma
  In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...
  
  

Computational group theory

• Coset enumeration
  Coset enumeration
  In mathematics, coset enumeration is the problem of counting the cosets of a subgroup H of a group G given in terms of a presentation. As a by-product, one obtains a permutation representation for G on the cosets of H...
  
  
• Schreier's subgroup lemma
  Schreier's subgroup lemma
  Schreier's subgroup lemma is a theorem in group theory used in the Schreier–Sims algorithm and also for finding a presentation of a subgroup.-Definition:Suppose H is a subgroup of G, which is finitely generated with generating set S, that is, G = ....
  
  
• Schreier–Sims algorithm
• Todd–Coxeter algorithm

Applications

• Computer algebra system
  Computer algebra system
  A computer algebra system is a software program that facilitates symbolic mathematics. The core functionality of a CAS is manipulation of mathematical expressions in symbolic form.-Symbolic manipulations:...
  
  
• Cryptography
  Cryptography
  Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
  
  
  • Discrete logarithm
    Discrete logarithm
    In mathematics, specifically in abstract algebra and its applications, discrete logarithms are group-theoretic analogues of ordinary logarithms. In particular, an ordinary logarithm loga is a solution of the equation ax = b over the real or complex numbers...
    
    
  • Triple DES
    Triple DES
    In cryptography, Triple DES is the common name for the Triple Data Encryption Algorithm block cipher, which applies the Data Encryption Standard cipher algorithm three times to each data block....
    
    
  • Caesar cipher
    Caesar cipher
    In cryptography, a Caesar cipher, also known as a Caesar's cipher, the shift cipher, Caesar's code or Caesar shift, is one of the simplest and most widely known encryption techniques. It is a type of substitution cipher in which each letter in the plaintext is replaced by a letter some fixed number...
    
    
• Exponentiating by squaring
• Knapsack problem
  Knapsack problem
  The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the count of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as...
  
  
• Shor's algorithm
  Shor's algorithm
  Shor's algorithm, named after mathematician Peter Shor, is a quantum algorithm for integer factorization formulated in 1994...
  
  
• Standard Model
  Standard Model
  The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...
  
  

Famous problems

• Burnside's problem
  Burnside's problem
  The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group...
  
  
• Classification of finite simple groups
  Classification of finite simple groups
  In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
  
  
• Herzog-Schönheim conjecture
• Subset sum problem
• Whitehead problem
  Whitehead problem
  In group theory, a branch of abstract algebra, the Whitehead problem is the following question:Shelah proved that Whitehead's problem is undecidable within standard ZFC set theory.-Refinement:...
  
  
• Word problem for groups
  Word problem for groups
  In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...
  
  

Group theorists

• N. Abel
  Niels Henrik Abel
  Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
  
  
• M. Aschbacher
  Michael Aschbacher
  Michael George Aschbacher is an American mathematician best known for his work on finite groups. He was a leading figure in the completion of the classification of finite simple groups in the 1970s and 1980s. It later turned out that the classification was incomplete, because the case of quasithin...
  
  
• R. Baer
  Reinhold Baer
  Reinhold Baer was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings....
  
  
• R. Brauer
  Richard Brauer
  Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...
  
  
• W. Burnside
  William Burnside
  William Burnside was an English mathematician. He is known mostly as an early contributor to the theory of finite groups....
  
  
• R. Carter
  Roger Carter (mathematician)
  Roger W. Carter is an emeritus professor at the University of Warwick. He defined Carter subgroups and wrote the standard reference Simple Groups of Lie Type. He obtained his PhD in 1960 and his dissertation was entitled Some Contributions to the Theory of Finite Soluble Groups.-Publications:*R.W....
  
  
• A. Cauchy
  Augustin Louis Cauchy
  Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...
  
  
• A. Cayley
  Arthur Cayley
  Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
  
  
• J.H. Conway
  John Horton Conway
  John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
  
  
• R. Dedekind
  Richard Dedekind
  Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
  
  
• L.E. Dickson
  Leonard Eugene Dickson
  Leonard Eugene Dickson was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remembered for a three-volume history of number theory.-Life:Dickson considered himself a Texan by...
  
  

• M. Dunwoody
  Martin Dunwoody
  Martin John Dunwoody is an Emeritus Professor of mathematics at the University of Southampton, England.He earned his Ph.D. in 1964 from the Australian National University. He held positions at the University of Sussex before becoming full Professor at the University of Southampton in 1992...
  
  
• W. Feit
  Walter Feit
  Walter Feit was a Jewish Austrian-American mathematician who worked in finite group theory and representation theory....
  
  
• B. Fischer
  Bernd Fischer
  Bernd Fischer may refer to:* Bernd Fischer , German mathematician.* Bernd Jürgen Fischer, historian and professor of history at Indiana University-Purdue University Fort Wayne....
  
  
• H. Fitting
  Hans Fitting
  Hans Fitting was a mathematician who worked in group theory...
  
  
• G. Frattini
  Giovanni Frattini
  Giovanni Frattini was an Italian mathematician, noted for his contributions to group theory.He entered the University of Rome in 1869, where he studied mathematics with Giuseppe Battaglini, Eugenio Beltrami, and Luigi Cremona, obtaining his PhD. in 1875.In 1885 he published a paper where he...
  
  
• G. Frobenius
  Ferdinand Georg Frobenius
  Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...
  
  
• E. Galois
  Évariste Galois
  Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
  
  
• G. Glauberman
  George Glauberman
  George Glauberman is a mathematician at the University of Chicago who works on finite simple groups. He proved the ZJ theorem and the Z* theorem....
  
  
• D. Gorenstein
  Daniel Gorenstein
  Daniel E. Gorenstein was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertation Gorenstein rings...
  
  
• R.L. Griess
  R. L. Griess
  Robert Louis Griess, Jr. is a mathematician working on finite simple groups and vertex algebras. He constructed the monster group using the Griess algebra. He received the AMS Leroy P...
  
  
• M. Hall, Jr.
  Marshall Hall (mathematician)
  Marshall Hall, Jr. was an American mathematician who made contributions to group theory and combinatorics.- Career :...
  
  

• P. Hall
  Philip Hall
  Philip Hall FRS , was an English mathematician.His major work was on group theory, notably on finite groups and solvable groups.-Biography:...
  
  
• G. Higman
  Graham Higman
  Graham Higman FRS was a leading British mathematician. He is known for his contributions to group theory....
  
  
• D. Hilbert
  David Hilbert
  David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
  
  
• O. Hölder
• B. Huppert
  Bertram Huppert
  Bertram Huppert is a German mathematician specializing in group theory and the representation theory of finite groups. His Endliche Gruppen is an influential textbook in group theory, and he has over 50 doctoral descendants.-Education:Bertram Huppert went to school in Bonn from 1934 until 1945...
  
  
• K. Iwasawa
  Kenkichi Iwasawa
  Kenkichi Iwasawa was a Japanese mathematician who is known for his influence on algebraic number theory.Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gunma Prefecture...
  
  
• Z. Janko
  Zvonimir Janko
  Zvonimir Janko is a Croatian mathematician who is the eponym of the Janko groups, sporadic simple groups in group theory.Janko was born in Bjelovar, Croatia. He studied at the University of Zagreb where he received Ph.D. in 1960. He then taught physics at a high school in Široki Brijeg in Bosnia...
  
  
• C. Jordan
  Camille Jordan
  Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...
  
  
• F. Klein
  Felix Klein
  Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
  
  
• A. Kurosh
  Aleksandr Gennadievich Kurosh
  Aleksandr Gennadievich Kurosh was a Soviet mathematician, known for his work in abstract algebra. He is credited with writing the first modern and high-level text on group theory, his The Theory of Groups published in 1944.He was born in Yartsevo near Smolensk, and died in Moscow. He received his...
  
  
• J.L. Lagrange

• C. Leedham-Green
  Charles Leedham-Green
  Charles R. Leedham-Green is a retired professor of mathematics at Queen Mary, University of London, known for his work in group theory. He completed his DPhil at the University of Oxford....
  
  
• F.W. Levi
  Friedrich Wilhelm Levi
  Friedrich Wilhelm Daniel Levi was a German mathematician known for his work in abstract algebra. He also worked in geometry, topology, set theory, and analysis...
  
  
• Sophus Lie
  Sophus Lie
  Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
  
  
• W. Magnus
  Wilhelm Magnus
  Wilhelm Magnus was a mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations....
  
  
• E. Mathieu
  Émile Léonard Mathieu
  Émile Léonard Mathieu was a French mathematician. He is most famous for his work in group theory and mathematical physics. He has given his name to the Mathieu functions, Mathieu groups and Mathieu transformation...
  
  
• G.A. Miller
  George Abram Miller
  George Abram Miller was an early group theorist whose many papers and texts were considered important by his contemporaries, but are now mostly considered only of historical importance...
  
  
• B.H. Neumann
  Bernhard Neumann
  Bernhard Hermann Neumann AC FRS was a German-born British mathematician who was one of the leading figures in group theory, greatly influencing the direction of the subject....
  
  
• H. Neumann
  Hanna Neumann
  Johanna Neumann was a German-born mathematician who worked on group theory.Johanna was born in Lankwitz, Steglitz-Zehlendorf, Germany. She attended Auguste-Viktoria-Schule and the University of Berlin and completed her studies in 1936 with distinctions in mathematics and physics. She began...
  
  
• J. Nielson
  Jakob Nielsen (mathematician)
  Jakob Nielsen was a Danish mathematician known for his work on automorphisms of surfaces. He was born in the village Mjels on the island of Als in North Schleswig, in modern day Denmark. His mother died when he was 3, and in 1900 he went to live with his aunt and was enrolled in the Realgymnasium...
  
  
• Emmy Noether
  Emmy Noether
  Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
  
  
• Ø. Ore
  Øystein Ore
  Øystein Ore was a Norwegian mathematician.-Life:Ore was graduated from the University of Oslo in 1922, with a Cand.Scient. degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled Zur Theorie der algebraischen Körper, supervised by Thoralf Skolem...
  
  

• O. Schreier
  Otto Schreier
  Otto Schreier was an Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups. He studied mathematics at the University of Vienna and obtained his doctorate in 1923, under the supervision of Philipp Furtwängler...
  
  
• I. Schur
  Issai Schur
  Issai Schur was a mathematician who worked in Germany for most of his life. He studied at Berlin...
  
  
• R. Steinberg
  Robert Steinberg
  Robert Steinberg is a mathematician at the University of California, Los Angeles who invented the Steinberg representation, the Steinberg group in algebraic K-theory, and the Steinberg groups in Lie theory that yield finite simple groups over finite fields. He received his Ph.D...
  
  
• M. Suzuki
  Michio Suzuki
  was a Japanese mathematician who studied group theory.-Biography:He was a Professor at the University of Illinois at Urbana-Champaign from 1953 to his death. He also had visiting positions at the University of Chicago , the Institute for Advanced Study , the University of Tokyo , and the...
  
  
• L. Sylow
  Peter Ludwig Mejdell Sylow
  Peter Ludwig Mejdell Sylow was a Norwegianmathematician, who proved foundational results in group theory. He was born and died in Christiania ....
  
  
• J. Thompson
  John G. Thompson
  John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize....
  
  
• J. Tits
  Jacques Tits
  Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
  
  
• Helmut Wielandt
  Helmut Wielandt
  thumb|right|Helmut WielandtHelmut Wielandt was a German mathematician who worked on permutation groups.- References:...
  
  
• H. Zassenhaus
  Hans Julius Zassenhaus
  Hans Julius Zassenhaus was a German mathematician, known for work in many parts of abstract algebra, and as a pioneer of computer algebra....
  
  
• M. Zorn
  Max August Zorn
  Max August Zorn was a German-born American mathematician. He was an algebraist, group theorist, and numerical analyst. He is best known for Zorn's lemma, a powerful tool in set theory that is applicable to a wide range of mathematical constructs such as vector spaces, ordered sets, etc...
  
  

See also

• List of abstract algebra topics
• List of category theory topics
• List of Lie group topics