# Group (mathematics)

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy four conditions called the group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group. However, the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: it 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. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity.

The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—a very active mathematical discipline—studies groups in their own right.a 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 addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups announced in 1983.aa Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.

Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• Group homomorphism
• (Semi-)direct product
• group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• List of group theory topics
• Glossary of group theory
Finite groups
• Classification of finite simple groups
• Cyclic group
• Zn
• Symmetric group
• Sn
• Dihedral group
• Dn
• Alternating group
• An
• Mathieu groups
• M11
• M12
• M22
• M23
• M24
• Conway groups
• Co1
• Co2
• Co3
• Janko groups
• J1
• J2
• J3
• J4
• Fischer groups
• F22
• F23
• F24
• Baby Monster group
• B
• Monster group
• M
Discrete groups and lattices
• Integers
• Z
• Lattice
• Modular groups
• PSL(2,Z)
• SL(2,Z)
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
• Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Elliptic curve
• Linear algebraic group
• Abelian variety
Algebraic structures
Group-like structures Semigroup and Monoid
Quasigroup and Loop
Abelian group
Ring-like structures Semiring
Near-ring
Ring
Commutative ring
Integral domain
Field
Lattice-like structures Semilattice
Lattice
Map of lattices
Module-like structures Group with operators
Module
Vector space
Algebra-like structures Algebra
Associative algebra
Non-associative algebra