1 Motivation
When we proved Lagrange's theorem in the previous chapter, we saw that the left cosets of a subgroup H partition the group G. A natural question arises: can we turn this partition into a group?
If we try to define a "product" of cosets by aH⋅bH=(ab)H, we immediately encounter a problem. If aH=a′H and bH=b′H, it is not automatically true that (ab)H=(a′b′)H. This product is well-defined precisely when H is a normal subgroup.
2 Definition of Normal Subgroups
A subgroup N of a group G is called a normal subgroup if gN=Ng for every g∈G. We write N⊴G. If N={e} and N=G, we say N is a proper normal subgroup and write N◃G.
3 Equivalent Conditions for Normality
Let G be a group and N≤G. The following are equivalent:
N⊴G (i.e. gN=Ng for all g∈G).
gNg−1=N for all g∈G.
gng−1∈N for all g∈G and n∈N.
gNg−1⊆N for all g∈G.
(1)⇒(2): Multiplying gN=Ng on the right by g−1 gives gNg−1=Ngg−1=N.(2)⇒(3): If gNg−1=N, then for any n∈N we have gng−1∈gNg−1=N.(3)⇒(4): Immediate.(4)⇒(1): If gNg−1⊆N for all g, then gN⊆Ng. Replacing g by g−1 gives g−1Ng⊆N, i.e. Ng⊆gN. Therefore gN=Ng. □
For g∈G and n∈N, the element gng−1 is called the conjugate of n by g. Condition (3) above says that N is closed under conjugation by every element ofG.
4 Examples of Normal Subgroups
For any group G, both {e}⊴G and G⊴G.
If G is abelian, then every subgroup of G is normal. Indeed, gng−1=gg−1n=n∈N for all g∈G and n∈N.
If g∈N, then gN=N=Ng trivially. If g∈/N, there are only two left cosets: N and gN. Since gN=N, we must have gN=G∖N. Similarly, the two right cosets are N and Ng=G∖N. Therefore gN=Ng. □
The alternating group An is a normal subgroup of Sn. This follows immediately from the preceding result, since [Sn:An]=2.
For A∈GLn(R) and B∈SLn(R), we have det(ABA−1)=det(A)det(B)det(A)−1=det(B)=1, so ABA−1∈SLn(R).
For any group homomorphism φ:G→G′, we have kerφ⊴G. Indeed, if n∈kerφ, then φ(gng−1)=φ(g)φ(n)φ(g)−1=φ(g)eG′φ(g)−1=eG′, so gng−1∈kerφ.
In S3, the subgroup H={e,(12)} is not normal. Indeed, (13)(12)(13)−1=(13)(12)(13)=(23)∈/H.
5 Construction of the Quotient Group
Let N⊴G. Define G/N={gN∣g∈G} (the set of all left cosets) with the operation (aN)(bN)=(ab)N.
Then G/N is a group, called the quotient group (or factor group) of G by N.
Well-definedness. Suppose aN=a′N and bN=b′N. Write a′=an1 and b′=bn2 with n1,n2∈N. Then a′b′=an1bn2=a(n1b)n2. Since N is normal, n1b=bn1′ for some n1′∈N. Hence a′b′=abn1′n2 with n1′n2∈N, so (a′b′)N=(ab)N.Associativity.(aN⋅bN)⋅cN=(ab)N⋅cN=((ab)c)N=(a(bc))N=aN⋅(bc)N=aN⋅(bN⋅cN).Identity. The identity element is eN=N. Indeed, aN⋅N=(ae)N=aN.Inverses. The inverse of aN is a−1N. Indeed, aN⋅a−1N=(aa−1)N=N. □
Since Z is abelian, nZ⊴Z. The quotient group Z/nZ is a cyclic group of order n, with elements 0ˉ,1ˉ,…,n−1 and the operation aˉ+bˉ=a+b.
We have A3={e,(123),(132)}⊴S3. The quotient group S3/A3 has order 2 and is isomorphic to Z/2Z. Its elements are A3 (the even permutations) and (12)A3 (the odd permutations).
Since SLn(R)⊴GLn(R), the quotient group is well-defined, and in fact GLn(R)/SLn(R)≅R× (via the determinant map).
6 The Canonical Projection
For N⊴G, the map π:G→G/N defined by π(g)=gN is called the natural homomorphism (or canonical projection).
The natural homomorphism π:G→G/N is a surjective homomorphism with kerπ=N.
π(ab)=(ab)N=(aN)(bN)=π(a)π(b), so π is a homomorphism. For any gN∈G/N, π(g)=gN, so π is surjective. Finally, kerπ={g∈G∣gN=N}={g∈G∣g∈N}=N. □
7 Homomorphisms and Normal Subgroups
Let φ:G→G′ be a group homomorphism. Then:
If H≤G, then φ(H)≤G′.
If H′≤G′, then φ−1(H′)≤G.
If N′⊴G′, then φ−1(N′)⊴G.
If φ is surjective and N⊴G, then φ(N)⊴G′.
Here φ−1(H′)={g∈G∣φ(g)∈H′} denotes the preimage.
(1) For a,b∈H, φ(a)φ(b)−1=φ(ab−1)∈φ(H).(2) For a,b∈φ−1(H′), φ(ab−1)=φ(a)φ(b)−1∈H′ (since H′≤G′), so ab−1∈φ−1(H′).(3) For g∈G and n∈φ−1(N′), φ(gng−1)=φ(g)φ(n)φ(g)−1∈N′ (since N′⊴G′), so gng−1∈φ−1(N′).(4) For g′∈G′ and φ(n)∈φ(N): since φ is surjective, g′=φ(g) for some g∈G. Then g′φ(n)g′−1=φ(g)φ(n)φ(g)−1=φ(gng−1)∈φ(N), using N⊴G. □
8 Simple Groups
A group G with ∣G∣>1 is called simple if it has no proper normal subgroups, i.e. the only normal subgroups of G are {e} and G itself.
Every group of prime order is simple.
By Lagrange's theorem, if ∣G∣=p (prime), the only possible subgroup orders are 1 and p. Hence the only subgroups are {e} and G, so G has no proper normal subgroups. □
For n≥5, the alternating group An is simple.
9 Analyzing Structure via Quotient Groups
Let N⊴G. There is an inclusion-preserving bijection between the subgroups of G that contain N and the subgroups of G/N: H↦H/N,Hˉ↦π−1(Hˉ),
where π:G→G/N is the canonical projection. Moreover, H⊴G if and only if H/N⊴G/N.
If N≤H≤G, then N⊴H (since N⊴G and N≤H), so H/N is a well-defined group, and H/N≤G/N.Conversely, if Hˉ≤G/N, then π−1(Hˉ)={g∈G∣gN∈Hˉ} is a subgroup of G containing N.One verifies that these two correspondences are mutual inverses and preserve both inclusion and normality. □
10 Summary and Next Steps
In this chapter we have covered:
The definition of normal subgroups and their equivalent characterizations (closure under conjugation).
Examples: abelian groups, subgroups of index 2, kernels, An, SLn.
Construction of the quotient group G/N and the proof of well-definedness.
The canonical projection π:G→G/N and the correspondence "normal subgroups = kernels."
Simple groups: definition and examples.
The correspondence theorem.
With normal subgroups and quotient groups in hand, we are ready to prove the central theorems of group theory — the isomorphism theorems (first, second, and third). These are variations on a single principle: "the image of a homomorphism is isomorphic to a quotient group," and they constitute the most powerful tools for analyzing the structure of groups.