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 operations ∗G, ∗H and identity elements
eG and eH, respectively) and let Φ:G→H be a group homomorphism
. Then, the kernel of Φ, i.e.
Ker(Φ)={g∈G:Φ(g)=eH}, |
is a normal subgroup of G.
Proof.
Let G,H and Φ be as in the statement of the lemma and let g∈G and k∈Ker(Φ). Then, Φ(k)=eH by definition and:
Φ(g∗Gk∗Gg-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-1∈G is also an element of the kernel of Φ. Since g∈G and k∈Ker(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:G→H, for some group H, such that the kernel of Φ is precisely K.
Proof.
Simply set H equal to the quotient group G/K and define Φ:G→G/K to be the natural projection
from G to G/K (i.e. Φ sends g∈G 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 subgroup is normal.
Example.
Let F be a field. Let us prove that the special linear group SL(n,F) is normal inside the general linear group
GL(n,F), for all n≥1. By the lemmas, it suffices to construct a homomorphism
of GL(n,F) with SL(n,F) as kernel. The determinant
of matrices is the homomorphism we are looking for. Indeed:
is a group homomorphism from to the multiplicative group and, by definition, the kernel is precisely , i.e. the matrices with determinant . Hence, is normal in .
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 |