1 Introduction
Given a group homomorphism φ:G→G′, what is the relationship between the kernel and the image of φ? The first isomorphism theorem provides a remarkably clean answer:
G/kerφ≅Imφ.
This theorem is the unifying principle of group theory. The second and third isomorphism theorems are variations on the same theme.
2 The First Isomorphism Theorem
Let φ:G→G′ be a group homomorphism. Then G/kerφ≅Imφ.
More precisely, the map φˉ:G/kerφ→Imφ defined by φˉ(gkerφ)=φ(g) is an isomorphism.
Set N=kerφ.Well-definedness. If gN=g′N, then g−1g′∈N=kerφ, so φ(g−1g′)=e, i.e. φ(g)=φ(g′). Hence φˉ(gN)=φˉ(g′N).Homomorphism.φˉ(gN⋅hN)=φˉ((gh)N)=φ(gh)=φ(g)φ(h)=φˉ(gN)φˉ(hN).Injectivity. If φˉ(gN)=e′, then φ(g)=e′, so g∈kerφ=N, and gN=N. Thus kerφˉ={N}.Surjectivity. For any φ(g)∈Imφ, we have φˉ(gN)=φ(g). □
3 Applications of the First Isomorphism Theorem
The determinant map det:GLn(R)→R× is a surjective homomorphism with ker(det)=SLn(R). By the first isomorphism theorem, GLn(R)/SLn(R)≅R×.
The sign map sgn:Sn→{+1,−1} is a surjective homomorphism with ker(sgn)=An. By the first isomorphism theorem, Sn/An≅Z/2Z.
The map exp:(R,+)→(R>0,×) defined by x↦ex is an isomorphism (a bijective homomorphism). We have ker(exp)={0}, and the first isomorphism theorem gives the tautology R/{0}≅R>0.
The map φ:Z→Z/nZ defined by φ(k)=kˉ is a surjective homomorphism with kerφ=nZ. The first isomorphism theorem gives Z/nZ≅Z/nZ— a tautology, but one that confirms the consistency of the notation.
Let G=G1×G2, and define π1:G→G1 by π1(g1,g2)=g1. This is a surjective homomorphism with kerπ1={e1}×G2. The first isomorphism theorem gives (G1×G2)/({e1}×G2)≅G1.
4 The Second Isomorphism Theorem (Diamond Isomorphism Theorem)
Let G be a group, H≤G, and N⊴G. Then:
HN={hn∣h∈H,n∈N} is a subgroup of G.
H∩N⊴H.
H/(H∩N)≅HN/N.
(1) For h1n1,h2n2∈HN, we have (h1n1)(h2n2)−1=h1n1n2−1h2−1. Since N⊴G, we can write n1n2−1h2−1=h2−1n′ for some n′∈N. Hence (h1n1)(h2n2)−1=h1h2−1n′∈HN.(2) Consider the homomorphism φ:H→HN/N defined by φ(h)=hN (this is well-defined since H⊆HN). Its kernel is kerφ={h∈H∣hN=N}={h∈H∣h∈N}=H∩N. Since kernels are normal, H∩N⊴H.(3) The map φ is surjective: for any hnN=hN∈HN/N, we have φ(h)=hN. By the first isomorphism theorem, H/(H∩N)=H/kerφ≅Imφ=HN/N. □
Let G=Z, H=mZ, and N=nZ. Then HN=gcd(m,n)Z and H∩N=lcm(m,n)Z, and the second isomorphism theorem gives mZ/lcm(m,n)Z≅gcd(m,n)Z/nZ.
5 The Third Isomorphism Theorem
Let G be a group with N⊴G and M⊴G, where N≤M. Then M/N⊴G/N and (G/N)/(M/N)≅G/M.
Define ψ:G/N→G/M by ψ(gN)=gM.Well-definedness. If gN=g′N, then g−1g′∈N≤M, so gM=g′M.Homomorphism.ψ(gN⋅hN)=ψ((gh)N)=(gh)M=gM⋅hM=ψ(gN)ψ(hN).Surjectivity. For any gM∈G/M, we have ψ(gN)=gM.Kernel.kerψ={gN∣gM=M}={gN∣g∈M}=M/N.By the first isomorphism theorem, (G/N)/(M/N)=(G/N)/kerψ≅Imψ=G/M. □
Let G=Z, N=6Z, and M=2Z. Since 6Z≤2Z≤Z, the third isomorphism theorem gives (Z/6Z)/(2Z/6Z)≅Z/2Z.
The left-hand side: Z/6Z={0ˉ,1ˉ,2ˉ,3ˉ,4ˉ,5ˉ} is divided by 2Z/6Z={0ˉ,2ˉ,4ˉ}, yielding a group of order 2. The right-hand side is also a group of order 2.
6 How to Use the Isomorphism Theorems
The isomorphism theorems are used in the following typical ways:
Identifying quotient groups. To answer "What is G/N isomorphic to?", construct a suitable homomorphism φ with kernel N, and apply the first isomorphism theorem.
Computing orders of subgroups. From the second isomorphism theorem, ∣HN/N∣=∣H/(H∩N)∣=∣H∣/∣H∩N∣, which gives the formula ∣HN∣=∣H∣⋅∣N∣/∣H∩N∣.
Simplifying iterated quotients. The third isomorphism theorem reduces a double quotient to a single one.
Let H,K≤G be finite, and let HK={hk∣h∈H,k∈K}. Then ∣HK∣=∣H∩K∣∣H∣⋅∣K∣.
When HK is a subgroup (e.g. when K⊴G), this follows directly from the second isomorphism theorem: ∣HK/K∣=∣H/(H∩K)∣ gives ∣HK∣/∣K∣=∣H∣/∣H∩K∣. In general, even when HK is not a subgroup, the same formula holds. Consider the map f:H×K→HK defined by f(h,k)=hk. Each element x∈HK has exactly ∣H∩K∣ preimages: if x=h1k1=h2k2, then h2−1h1=k2k1−1∈H∩K, so the fibre of f over x has the form {(h⋅t,t−1⋅k)∣t∈H∩K} for any fixed preimage (h,k). Thus ∣H∣⋅∣K∣=∣HK∣⋅∣H∩K∣. □
7 Summary
In this chapter we have established:
The first isomorphism theorem: G/kerφ≅Imφ.
The second isomorphism theorem: H/(H∩N)≅HN/N.
The third isomorphism theorem: (G/N)/(M/N)≅G/M.
Concrete calculations and applications of each theorem.
The product formula: ∣HK∣=∣H∣⋅∣K∣/∣H∩K∣.
These three isomorphism theorems are the fundamental tools for dissecting the structure of groups. The next step is the theory of group actions (orbits, stabilizers, and the class equation), which leads ultimately to the Sylow theorems.