homomorphic image of group

Theorem.  The homomorphic imagePlanetmathPlanetmathPlanetmath of a group is a group.  More detailed, if f is a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from the group  (G,)  to the groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath(Γ,),  then the groupoid  (f(G),)  also is a group.  Especially, the isomorphic image of a group is a group.

Proof.  Let α,β,γ be arbitrary elements of the image f(G) and a,b,c some elements of G such that  f(a)=α,f(b)=β,f(c)=γ.  Then


whence f(G) is closed underPlanetmathPlanetmath”, and we, in fact, can speak of a groupoid  (f(G),).

Secondly, we can calculate

(αβ)γ =(f(a)f(b))f(c)

whence the associativity is in in the groupoid (f(G),).

Let e be the identity elementMathworldPlanetmath of  (G,)  and  f(e)=ε.  Then


and therefore ε is an identity element in f(G).

If  f(a-1)=α, then


Thus any element α of f(G) has in f(G) an inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Accordingly,  (f(G),)  is a group.

Remark 1.  If  (G,)  is AbelianMathworldPlanetmath, the same is true for  (f(G),).

Remark 2.  Analogically, one may prove that the homomorphic image of a ring is a ring.

Example.  If we define the mapping f from the group  (,+)  to the groupoid  (9,)  by


then f is homomorphism:


The image f() consists of powers of the residue class (http://planetmath.org/CongruencesPlanetmathPlanetmathPlanetmathPlanetmath) 4, which are


These apparently form the cyclic groupMathworldPlanetmath of order 3.

Title homomorphic image of group
Canonical name HomomorphicImageOfGroup
Date of creation 2013-03-22 18:56:27
Last modified on 2013-03-22 18:56:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Theorem
Classification msc 20A05
Classification msc 08A05
Related topic GroupHomomorphism
Related topic CorrespondenceBetweenNormalSubgroupsAndHomomorphicImages