Group homomorphism

Tablecontents

1 Definition 2 ImageKernel 3 Examples 4 The categorygroups 5 Isomorphisms, EndomorphismsAutomorphisms 6 Homomorphismsabelian groups

Definition

Given two groups (G, *)(H, ·),group homomorphism from (G, *)(H, ·) isfunction h : G -> H such thatall uvGholds that

h(u * v) = h(u) · h(v)

From this property, one can deduce that h mapsidentity element e_GG toidentity element e_HH,it also maps inversesinverses insense that h(u^-1) = h(u)^-1. Hence one can say that h "is compatible withgroup structure".

Older notations forhomomorphism h(x) may be x_h, though this may be confused as an index orgeneral subscript.

ImageKernel

We definekernelhbe

ker(h) = { uG : h(u) = e_H }

andimagehbe

im(h) = { h(u) : uG }.

The kernel isnormal subgroupG (in fact, h(g^-1 u g) = h(g)^-1 e_H h(g) = h(g)^-1 h(g) = e_H) andimage issubgroupH. The homomorphism hinjective (and calledgroup monomorphism) ifonly if ker(h) = {e_G}.

Examples

• Considercyclic group Z/3Z = {0, 1, 2} andgroupintegers Zaddition. The map h : Z -> Z/3Zh(u) = u modulo 3 isgroup homorphism (see modular arithmetic). Itsurjectiveits kernel consistsall integers whichdivisible by 3.

• The exponential map yieldsgroup homorphism fromgroupreal numbers Raddition togroupnon-zero real numbers R^*multiplication. The kernel{0} andimage consists ofpositive real numbers.

• The exponential map also yieldsgroup homomorphism fromgroupcomplex numbers Caddition togroupnon-zero complex numbers C^*multiplication. This mapsurjectivehaskernel { 2πki : kZ }, as can be seen from Euler's formula.

• Given any two groups GH,map h : G -> H which sends every elementG toidentity elementH ishomomorphism; its kernelallG.

• Given any group G,identity map id : G -> Gid(u) = uall uG isgroup homomorphism.

The categorygroups

If h : G -> Hk : H -> Kgroup homomorphisms, then sok o h : G -> K. This shows thatclassall groups, togethergroup homomorphisms as morphisms, formscategory.

Isomorphisms, EndomorphismsAutomorphisms

Ifhomomorphism h isbijection, then one can show that its inversealsogroup homomorphism,hcalled group isomorphism;this case,groups GHcalled isomorphic:differ only innotationtheir elements andidenticalall practical purposes.

If h: G -> G isgroup homomorphism, we callan endomorphismG. If furthermore itbijectivehence an isomorphism, itcalled an automorphism. The setall automorphisms ofgroup G,functional composition as operation, forms itselfgroup,automorphism groupG. Itdenoted by Aut(\G). As an example,automorphism group(Z, +) contains only two elements,identitymultiplication-1; itisomorphicZ/2Z.

Homomorphismsabelian groups

If GHabelian (i.e. commutative) groups, thenset Hom(G, H)all group homomorphisms from GHitself an abelian group:sum h + ktwo homomorphismsdefined by

(h + k)(u) = h(u) + k(u)   all uG.

The commutativityHneededprove that h + kagaingroup homomorphism. The additionhomomorphismscompatible withcompositionhomomorphisms infollowing sense: if fin Hom(K, G), h, kelementsHom(G, H),gin Hom(H,L), then

(h + k) o f = (h o f) + (k o f)     g o (h + k) = (g o h) + (g o k).

This shows thatset End(G)all endomorphismsan abelian group formsring,endomorphism ringG. For example,endomorphism ring ofabelian group consisting ofdirect sumtwo copiesZ/2Z (the Klein four-group)isomorphic toring2-by-2 matricesentriesZ/2Z. The above compatibility also shows thatcategoryall abelian groupsgroup homomorphisms formspreadditive category;existencedirect sumswell-behaved kernels makes this categoryprototypical examplean abelian category.