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单词 GroupHomomorphism
释义

group homomorphism


Let (G,) and (K,) be two groups. A group homomorphismMathworldPlanetmath is afunction ϕ:GK such thatϕ(st)=ϕ(s)ϕ(t) for all s,tG.

A compositionMathworldPlanetmath of group homomorphisms is again a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Let ϕ:GK a group homomorphism.Then the kernel of ϕ is a normal subgroupMathworldPlanetmath of G,and the image of ϕ is a subgroupMathworldPlanetmathPlanetmath of K.Also, ϕ(gn)=ϕ(g)n for all gG and for all n.In particular,taking n=-1 we have ϕ(g-1)=ϕ(g)-1 for all gG,and taking n=0 we have ϕ(1G)=1K,where 1G and 1K are the identity elementsMathworldPlanetmath of G and K,respectively.

Some special homomorphisms have special names.If the homomorphism ϕ:GK is injectivePlanetmathPlanetmath,we say that ϕ is a monomorphismMathworldPlanetmathPlanetmath,and if ϕ is surjectivePlanetmathPlanetmath we call it an epimorphismMathworldPlanetmath.When ϕ is both injective and surjective (that is, bijectiveMathworldPlanetmath) we call it an isomorphismMathworldPlanetmathPlanetmathPlanetmath.In the latter case we also say that G and K are isomorphic, meaning they are basically the same group (have the same structureMathworldPlanetmath).A homomorphism from G on itself is called an endomorphism,and if it is bijective then it is called an automorphism.

Titlegroup homomorphism
Canonical nameGroupHomomorphism
Date of creation2013-05-17 17:53:21
Last modified on2013-05-17 17:53:21
Owneryark (2760)
Last modified byunlord (1)
Numerical id27
Authoryark (1)
Entry typeDefinition
Classificationmsc 20A05
Synonymhomomorphism
Synonymhomomorphism of groups
Related topicGroup
Related topicKernel
Related topicSubgroup
Related topicTypesOfHomomorphisms
Related topicKernelOfAGroupHomomorphism
Related topicGroupActionsAndHomomorphisms
Related topicEndomorphism2
Related topicGroupsOfRealNumbers
Related topicHomomorphicImageOfGroup
Definesepimorphism
Definesmonomorphism
Definesautomorphism
Definesendomorphism
Definesisomorphism
Definesisomorphic
Definesgroup epimorphism
Definesgroup monomorphism
Definesgroup automorphism
Definesgroup endomorphismPlanetmathPlanetmath
Definesgroup isomorphism
Definesepimorphism of groups
Definesmonomorphism of groups
Definesautomorphism of a group
Definesendom
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更新时间:2025/5/4 16:17:46