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单词 Group
释义

group


Group.
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 ofthe 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 groupoperation of G.

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

It can be proved that there is only one identity element, and that forevery element there is only one inverse.  Because of this we usuallydenote the inverse of a as a-1 or -a when we are usingadditive notation.  The identity element is also called neutralelement due to its behavior with respect to the operation, and thusa-1 is sometimes (although uncommonly) called the neutralizing element of a. An element of a group besides theidentity 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.

Titlegroup
Canonical nameGroup
Date of creation2013-03-22 11:42:53
Last modified on2013-03-22 11:42:53
Ownerdrini (3)
Last modified bydrini (3)
Numerical id34
Authordrini (3)
Entry typeDefinition
Classificationmsc 14F99
Classificationmsc 08A99
Classificationmsc 20A05
Classificationmsc 20-00
Classificationmsc 83C99
Classificationmsc 32C05
Related topicSubgroupMathworldPlanetmathPlanetmath
Related topicCyclicGroup
Related topicSimple
Related topicSymmetricGroup
Related topicFreeGroup
Related topicRing
Related topicField
Related topicGroupHomomorphism
Related topicLagrangesTheorem
Related topicIdentityElement
Related topicProperSubgroup
Related topicGroupoidPlanetmathPlanetmathPlanetmathPlanetmath
Related topicFundamentalGroup
Related topicTopologicalGroup
Related topicLieGroup
Related topicProofThatGInGImpliesThatLangleGRangleLeG
Related topicGeneralizedCyclicGroup
DefinesidentityPlanetmathPlanetmath
Definesinverse
Definesneutralizing element
Definesnon-trivial element
Definesnontrivial element
Definesgroup operation
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更新时间:2025/5/4 14:43:53