A group is a pair (G,*), where G is a non-empty set and “*” is a binary operationMathworldPlanetmath on G, such that the following conditions hold:

  • For any a,b in G,  a*b  belongs to G. (The operationMathworldPlanetmath*” is closed).

  • For any  a,b,cG,  (a*b)*c=a*(b*c).  (Associativity of the operation).

  • There is an element eG such that  g*e=e*g=g  for any  gG. (Existence of identity elementMathworldPlanetmath).

  • For any  gG  there exists an element h such that  g*h=h*g=e.  (Existence of inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath).

If G is a group under *, then * is referred to as the group operation of G.

Usually, the symbol “*” is omitted and we write  ab  for a*b.  Sometimes, the symbol “+” is used to represent the operation, especially when the group is abelianMathworldPlanetmath.

It can be proved that there is only one identity element, and that for every element there is only one inverse.  Because of this we usually denote the inverse of a as a-1 or -a when we are using additive notation.  The identity element is also called neutral element due to its behavior with respect to the operation, and thus a-1 is sometimes (although uncommonly) called the neutralizing element of a. An element of a group besides the identity element is sometimes called a non-trivial element.

Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actionsMathworldPlanetmath.  In fact, much of the study of groups themselves is conducted using group actions.

Title group
Canonical name Group
Date of creation 2013-03-22 11:42:53
Last modified on 2013-03-22 11:42:53
Owner drini (3)
Last modified by drini (3)
Numerical id 34
Author drini (3)
Entry type Definition
Classification msc 14F99
Classification msc 08A99
Classification msc 20A05
Classification msc 20-00
Classification msc 83C99
Classification msc 32C05
Related topic SubgroupMathworldPlanetmathPlanetmath
Related topic CyclicGroup
Related topic Simple
Related topic SymmetricGroup
Related topic FreeGroup
Related topic Ring
Related topic Field
Related topic GroupHomomorphism
Related topic LagrangesTheorem
Related topic IdentityElement
Related topic ProperSubgroup
Related topic GroupoidPlanetmathPlanetmathPlanetmathPlanetmath
Related topic FundamentalGroup
Related topic TopologicalGroup
Related topic LieGroup
Related topic ProofThatGInGImpliesThatLangleGRangleLeG
Related topic GeneralizedCyclicGroup
Defines identityPlanetmathPlanetmath
Defines inverse
Defines neutralizing element
Defines non-trivial element
Defines nontrivial element
Defines group operation