the kernel of a group homomorphism is a normal subgroup

In this entry we show the following simple lemma:

Lemma 1.

Let G and H be groups (with group operationsMathworldPlanetmath G, H and identity elementsMathworldPlanetmath eG and eH, respectively) and let Φ:GH be a group homomorphismMathworldPlanetmath. Then, the kernel of Φ, i.e.


is a normal subgroupMathworldPlanetmath of G.


Let G,H and Φ be as in the statement of the lemma and let gG and kKer(Φ). Then, Φ(k)=eH by definition and:

Φ(gGkGg-1) = Φ(g)HΦ(k)HΦ(g-1)
= Φ(g)H(eH)HΦ(g-1)
= Φ(g)HΦ(g-1)
= Φ(g)HΦ(g)-1
= eH,

where we have used several times the properties of group homomorphisms and the properties of the identity element eH. Thus, Φ(gkg-1)=eH and gkg-1G is also an element of the kernel of Φ. Since gG and kKer(Φ) were arbitrary, it follows that Ker(Φ) is normal in G. ∎

Lemma 2.

Let G be a group and let K be a normal subgroup of G. Then there exists a group homomorphism Φ:GH, for some group H, such that the kernel of Φ is precisely K.


Simply set H equal to the quotient groupMathworldPlanetmath G/K and define Φ:GG/K to be the natural projectionMathworldPlanetmath from G to G/K (i.e. Φ sends gG to the coset gK). Then it is clear that the kernel of Φ is precisely formed by those elements of K. ∎

Although the first lemma is very simple, it is very useful when one tries to prove that a subgroupMathworldPlanetmathPlanetmath is normal.


Let F be a field. Let us prove that the special linear groupMathworldPlanetmath SL(n,F) is normal inside the general linear groupMathworldPlanetmath GL(n,F), for all n1. By the lemmas, it suffices to construct a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of GL(n,F) with SL(n,F) as kernel. The determinantMathworldPlanetmath of matrices is the homomorphism we are looking for. Indeed:


is a group homomorphism from GL(n,F) to the multiplicative groupMathworldPlanetmath F× and, by definition, the kernel is precisely SL(n,F), i.e. the matrices with determinant =1. Hence, SL(n,F) is normal in GL(n,F).

Title the kernel of a group homomorphism is a normal subgroup
Canonical name TheKernelOfAGroupHomomorphismIsANormalSubgroup
Date of creation 2013-03-22 17:20:34
Last modified on 2013-03-22 17:20:34
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 8
Author alozano (2414)
Entry type Theorem
Classification msc 20A05
Related topic KernelOfAGroupHomomorphism
Related topic NaturalProjection