group homomorphism

Let (G,) and (K,) be two groups. A group homomorphismMathworldPlanetmath is a function ϕ:GK such that ϕ(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.

Title group homomorphism
Canonical name GroupHomomorphism
Date of creation 2013-05-17 17:53:21
Last modified on 2013-05-17 17:53:21
Owner yark (2760)
Last modified by unlord (1)
Numerical id 27
Author yark (1)
Entry type Definition
Classification msc 20A05
Synonym homomorphism
Synonym homomorphism of groups
Related topic Group
Related topic Kernel
Related topic Subgroup
Related topic TypesOfHomomorphisms
Related topic KernelOfAGroupHomomorphism
Related topic GroupActionsAndHomomorphisms
Related topic Endomorphism2
Related topic GroupsOfRealNumbers
Related topic HomomorphicImageOfGroup
Defines epimorphism
Defines monomorphism
Defines automorphism
Defines endomorphism
Defines isomorphism
Defines isomorphic
Defines group epimorphism
Defines group monomorphism
Defines group automorphism
Defines group endomorphismPlanetmathPlanetmath
Defines group isomorphism
Defines epimorphism of groups
Defines monomorphism of groups
Defines automorphism of a group
Defines endom