AATA The Isomorphism Theorems

Section 11.2 The Isomorphism Theorems

Although it is not evident at first, factor groups correspond exactly to homomorphic images, and we can use factor groups to study homomorphisms. We already know that with every group homomorphism

ϕ : G \to H

we can associate a normal subgroup of

G,

\ker ϕ .

The converse is also true; that is, every normal subgroup of a group

G

gives rise to homomorphism of groups.

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Let

H

be a normal subgroup of

G .

Define the natural or canonical homomorphism

ϕ : G \to G / H

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ϕ (g) = g H .

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This is indeed a homomorphism, since

ϕ (g_{1} g_{2}) = g_{1} g_{2} H = g_{1} H g_{2} H = ϕ (g_{1}) ϕ (g_{2}) .

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The kernel of this homomorphism is

H .

The following theorems describe the relationships between group homomorphisms, normal subgroups, and factor groups.

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Theorem 11.10. First Isomorphism Theorem.

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ψ : G \to H

is a group homomorphism with

K = \ker ψ,

then

K

is normal in

G .

Let

ϕ : G \to G / K

be the canonical homomorphism. Then there exists a unique isomorphism

η : G / K \to ψ (G)

such that

ψ = η ϕ .

Proof.

We already know that

K

is normal in

G .

Define

η : G / K \to ψ (G)

η (g K) = ψ (g) .

We first show that

η

is a well-defined map. If

g_{1} K = g_{2} K,

then for some

k \in K,

g_{1} k = g_{2};

consequently,

η (g_{1} K) = ψ (g_{1}) = ψ (g_{1}) ψ (k) = ψ (g_{1} k) = ψ (g_{2}) = η (g_{2} K) .

Thus,

η

does not depend on the choice of coset representatives and the map

η : G / K \to ψ (G)

is uniquely defined since

ψ = η ϕ .

We must also show that

η

is a homomorphism. Indeed,

\begin{aligned} η (g_{1} K g_{2} K) & = η (g_{1} g_{2} K) \\ = ψ (g_{1} g_{2}) \\ = ψ (g_{1}) ψ (g_{2}) \\ = η (g_{1} K) η (g_{2} K) . \end{aligned}

Clearly,

η

is onto

ψ (G) .

To show that

η

is one-to-one, suppose that

η (g_{1} K) = η (g_{2} K) .

Then

ψ (g_{1}) = ψ (g_{2}) .

This implies that

ψ (g_{1}^{- 1} g_{2}) = e,

g_{1}^{- 1} g_{2}

is in the kernel of

ψ;

hence,

g_{1}^{- 1} g_{2} K = K;

that is,

g_{1} K = g_{2} K .

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Mathematicians often use diagrams called commutative diagrams to describe such theorems. The following diagram “commutes” since

ψ = η ϕ .

$A diagram with G mapped with an arrow to H by psi on the top row. G gets mapped with an arrow down and right to G/K by phi. G/K gets mapped with an arrow up and right to H by eta.$

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Example 11.11.

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Let

G

be a cyclic group with generator

g .

Define a map

ϕ : Z \to G

n \mapsto g^{n} .

This map is a surjective homomorphism since

ϕ (m + n) = g^{m + n} = g^{m} g^{n} = ϕ (m) ϕ (n) .

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Clearly

ϕ

is onto. If

| g | = m,

then

g^{m} = e .

Hence,

\ker ϕ = m Z

and

Z / \ker ϕ = Z / m Z ≅ G .

On the other hand, if the order of

g

is infinite, then

\ker ϕ = 0

and

ϕ

is an isomorphism of

G

and

Z .

Hence, two cyclic groups are isomorphic exactly when they have the same order. Up to isomorphism, the only cyclic groups are

Z

and

Z_{n} .

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Theorem 11.12. Second Isomorphism Theorem.

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Let

H

be a subgroup of a group

G

(not necessarily normal in

G

) and

N

a normal subgroup of

G .

Then

H N

is a subgroup of

G,

H \cap N

is a normal subgroup of

H,

and

H / H \cap N ≅ H N / N .

Proof.

We will first show that

H N = {h n : h \in H, n \in N}

is a subgroup of

G .

Suppose that

h_{1} n_{1}, h_{2} n_{2} \in H N .

Since

N

is normal,

(h_{2})^{- 1} n_{1} h_{2} \in N .

(h_{1} n_{1}) (h_{2} n_{2}) = h_{1} h_{2} ((h_{2})^{- 1} n_{1} h_{2}) n_{2}

is in

H N .

The inverse of

h n \in H N

is in

H N

since

(h n)^{- 1} = n^{- 1} h^{- 1} = h^{- 1} (h n^{- 1} h^{- 1}) .

Next, we prove that

H \cap N

is normal in

H .

Let

h \in H

and

n \in H \cap N .

Then

h^{- 1} n h \in H

since each element is in

H .

Also,

h^{- 1} n h \in N

since

N

is normal in

G;

therefore,

h^{- 1} n h \in H \cap N .

Now define a map

ϕ

from

H

H N / N

h \mapsto h N .

The map

ϕ

is onto, since any coset

h n N = h N

is the image of

h

H .

We also know that

ϕ

is a homomorphism because

ϕ (h h^{'}) = h h^{'} N = h N h^{'} N = ϕ (h) ϕ (h^{'}) .

By the First Isomorphism Theorem, the image of

ϕ

is isomorphic to

H / \ker ϕ;

that is,

H N / N = ϕ (H) ≅ H / \ker ϕ .

Since

\ker ϕ = {h \in H : h \in N} = H \cap N,

H N / N = ϕ (H) ≅ H / H \cap N .

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Theorem 11.13. Correspondence Theorem.

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Let

N

be a normal subgroup of a group

G .

Then

H \mapsto H / N

is a one-to-one correspondence between the set of subgroups

H

G

containing

N

and the set of subgroups of

G / N .

Furthermore, the normal subgroups of

G

containing

N

correspond to normal subgroups of

G / N .

Proof.

Let

H

be a subgroup of

G

containing

N .

Since

N

is normal in

H,

H / N

is a factor group. Let

a N

and

b N

be elements of

H / N .

Then

(a N) (b^{- 1} N) = a b^{- 1} N \in H / N;

hence,

H / N

is a subgroup of

G / N .

Let

S

be a subgroup of

G / N .

This subgroup is a set of cosets of

N .

H = {g \in G : g N \in S},

then for

h_{1}, h_{2} \in H,

we have that

(h_{1} N) (h_{2} N) = h_{1} h_{2} N \in S

and

h_{1}^{- 1} N \in S .

Therefore,

H

must be a subgroup of

G .

Clearly,

H

contains

N .

Therefore,

S = H / N .

Consequently, the map

H \mapsto H / N

is onto.

Suppose that

H_{1}

and

H_{2}

are subgroups of

G

containing

N

such that

H_{1} / N = H_{2} / N .

h_{1} \in H_{1},

then

h_{1} N \in H_{1} / N .

Hence,

h_{1} N = h_{2} N \subset H_{2}

for some

h_{2}

H_{2} .

However, since

N

is contained in

H_{2},

we know that

h_{1} \in H_{2}

H_{1} \subset H_{2} .

Similarly,

H_{2} \subset H_{1} .

Since

H_{1} = H_{2},

the map

H \mapsto H / N

is one-to-one.

Suppose that

H

is normal in

G

and

N

is a subgroup of

H .

Then it is easy to verify that the map

G / N \to G / H

defined by

g N \mapsto g H

is a homomorphism. The kernel of this homomorphism is

H / N,

which proves that

H / N

is normal in

G / N .

Conversely, suppose that

H / N

is normal in

G / N .

The homomorphism given by

G \to G / N \to \frac{G / N}{H / N}

has kernel

H .

Hence,

H

must be normal in

G .

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Notice that in the course of the proof of Theorem 11.13, we have also proved the following theorem.

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Theorem 11.14. Third Isomorphism Theorem.

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Let

G

be a group and

N

and

H

be normal subgroups of

G

with

N \subset H .

Then

G / H ≅ \frac{G / N}{H / N} .

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Example 11.15.

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By the Third Isomorphism Theorem,

Z / m Z ≅ (Z / m n Z) / (m Z / m n Z) .

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Since

| Z / m n Z | = m n

and

| Z / m Z | = m,

we have

| m Z / m n Z | = n .

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