# Group (mathematics)

**groupgroupsgroup operationgroup structuregroup lawmathematical groupgroup theoryelementary group theorygroup axiomsabstract group**

In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility.wikipedia

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### Lie group

**Lie groupsLie subgroupmatrix Lie group**

Lie groups are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and point groups are used to help understand symmetry phenomena in molecular chemistry.

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are smooth.

### Symmetry group

**symmetrypoint group symmetrysymmetry groups**

For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other.

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition.

### Point group

**point groupssymmetryRosette groups**

Lie groups are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and point groups are used to help understand symmetry phenomena in molecular chemistry.

In geometry, a point group is a group of geometric symmetries (isometries) that keep at least one point fixed.

### Poincaré group

**Poincaré symmetryPoincaré algebraPoincaré transformation**

Lie groups are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and point groups are used to help understand symmetry phenomena in molecular chemistry.

The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries.

### Closure (mathematics)

**closedclosureclosed under**

In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility.

For example, one may define a group as a set with a binary product operator obeying several axioms, including an axiom that the product of any two elements of the group is again an element.

### Group theory

**groupgroup theoreticalgroup-theoretic**

Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups.

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

### Quotient group

**quotientfactor groupquotients**

To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups.

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out).

### Simple group

**simplesimple groupssimplicity**

To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups.

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

### Finite group

**finitefinite group theoryfinite groups**

A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004.

In abstract algebra, a finite group is a group, of which the underlying set contains a finite number of elements.

### Group representation

**representationrepresentationsrepresentation theory**

In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory.

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations (i.e. automorphisms) of vector spaces; in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication.

### Subgroup

**subgroupsproper subgroupgroup-subgroup**

To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups.

In group theory, a branch of mathematics, 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 ∗.

### Classification of finite simple groups

**classification program for finite simple groupsClassification theoremclassified**

A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004.

These groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime numbers are the basic building blocks of the natural numbers.

### Representation theory

**linear representationrepresentationsrepresentation**

In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory.

The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras.

### Finitely generated group

**finitely generatedfinitely-generated groupfinitely generated subgroup**

Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements.

### Abelian group

**abelianabelian groupsadditive group**

Groups for which the commutativity equation a • b = b • a always holds are called abelian groups (in honor of Niels Henrik Abel).

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 the order in which they are written.

### Computational group theory

**computationcomputationalcomputational investigations**

In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory.

groups by means of computers.

### Topological group

**continuous groupclosed subgrouptopological groups**

It is also useful for talking of properties of the inverse operation, as needed for defining topological groups and group objects.

In mathematics, a topological group is a group G together with a topology on G such that both the group's binary operation and the function mapping group elements to their respective inverses are continuous functions with respect to the topology.

### Galois group

**Galois groupsautomorphism groupgroups**

The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term of group (groupe, in French) for the symmetry group of the roots of an equation, now called a Galois group.

In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.

### Symmetry

**symmetricalsymmetricsymmetries**

Groups share a fundamental kinship with the notion of symmetry.

The set of operations that preserve a given property of the object form a group.

### Axiom

**axiomspostulateaxiomatic**

In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility.

Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory, group theory, topology, vector spaces) without any particular application in mind.

### Dihedral group

**dihedral symmetrydihedralDih 2**

These symmetries determine a group called the dihedral group of degree 4, denoted

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

### Group object

It is also useful for talking of properties of the inverse operation, as needed for defining topological groups and group objects.

In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets.

### Molecular symmetry

**symmetrypoint grouporbital symmetry**

The symmetry operations of a molecule (or other object) form a group.

### Binary operation

**binary operatoroperationbinary**

In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility.

Binary operations are the keystone of most algebraic structures, that are studied in algebra, and used in all mathematics, such as fields, groups, monoids, rings, algebras, and many more.

### Rotation (mathematics)

**rotationrotationsrotate**

All rotations about a fixed point form a group under composition called the rotation group (of a particular space).