AATA The Isomorphism Theorems

Section 4.3 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 $\phi: G \rightarrow H$ we can associate a normal subgroup of $G\text{,}$ $\ker \phi\text{.}$ The converse is also true; that is, every normal subgroup of a group $G$ gives rise to homomorphism of groups.

Let $H$ be a normal subgroup of $G\text{.}$ Define the natural or canonical homomorphism

\begin{equation*} \phi : G \rightarrow G/H \end{equation*}

\begin{equation*} \phi(g) = gH\text{.} \end{equation*}

This is indeed a homomorphism, since

\begin{equation*} \phi( g_1 g_2 ) = g_1 g_2 H = g_1 H g_2 H = \phi( g_1) \phi( g_2 )\text{.} \end{equation*}

The kernel of this homomorphism is $H\text{.}$ The following theorems describe the relationships between group homomorphisms, normal subgroups, and factor groups.

Theorem 4.22. First Isomorphism Theorem.

If $\psi : G \rightarrow H$ is a group homomorphism with $K =\ker \psi\text{,}$ then $K$ is normal in $G\text{.}$ Let $\phi: G \rightarrow G/K$ be the canonical homomorphism. Then there exists a unique isomorphism $\eta: G/K \rightarrow \psi(G)$ such that $\psi = \eta \phi\text{.}$

Proof.

We already know that $K$ is normal in $G\text{.}$ Define $\eta: G/K \rightarrow \psi(G)$ by $\eta(gK) = \psi(g)\text{.}$ We first show that $\eta$ is a well-defined map. If $g_1 K =g_2 K\text{,}$ then for some $k \in K\text{,}$ $g_1 k=g_2\text{;}$ consequently,

\begin{equation*} \eta(g_1 K) = \psi(g_1) = \psi(g_1) \psi(k) = \psi(g_1k) = \psi(g_2) = \eta(g_2 K)\text{.} \end{equation*}

Thus, $\eta$ does not depend on the choice of coset representatives and the map $\eta: G/K \rightarrow \psi(G)$ is uniquely defined since $\psi = \eta \phi\text{.}$ We must also show that $\eta$ is a homomorphism. Indeed,

\begin{align*} \eta( g_1K g_2K ) & = \eta(g_1 g_2K)\\ & = \psi(g_1 g_2)\\ & = \psi(g_1) \psi(g_2)\\ & = \eta( g_1K) \eta( g_2K )\text{.} \end{align*}

Clearly, $\eta$ is onto $\psi( G)\text{.}$ To show that $\eta$ is one-to-one, suppose that $\eta(g_1 K) = \eta(g_2 K)\text{.}$ Then $\psi(g_1) = \psi(g_2)\text{.}$ This implies that $\psi( g_1^{-1} g_2 ) = e\text{,}$ or $g_1^{-1} g_2$ is in the kernel of $\psi\text{;}$ hence, $g_1^{-1} g_2K = K\text{;}$ that is, $g_1K =g_2K\text{.}$

Mathematicians often use diagrams called commutative diagrams to describe such theorems. The following diagram “commutes” since $\psi = \eta \phi\text{.}$

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

Example 4.23.

Let $G$ be a cyclic group with generator $g\text{.}$ Define a map $\phi : {\mathbb Z} \rightarrow G$ by $n \mapsto g^n\text{.}$ This map is a surjective homomorphism since

\begin{equation*} \phi( m + n) = g^{m+n} = g^m g^n = \phi(m) \phi(n)\text{.} \end{equation*}

Clearly $\phi$ is onto. If $|g| = m\text{,}$ then $g^m = e\text{.}$ Hence, $\ker \phi = m {\mathbb Z}$ and ${\mathbb Z} / \ker \phi = {\mathbb Z} / m {\mathbb Z} \cong G\text{.}$ On the other hand, if the order of $g$ is infinite, then $\ker \phi = 0$ and $\phi$ is an isomorphism of $G$ and ${\mathbb Z}\text{.}$ Hence, two cyclic groups are isomorphic exactly when they have the same order. Up to isomorphism, the only cyclic groups are ${\mathbb Z}$ and ${\mathbb Z}_n\text{.}$

Theorem 4.24. Second Isomorphism Theorem.

Let $H$ be a subgroup of a group $G$ (not necessarily normal in $G$) and $N$ a normal subgroup of $G\text{.}$ Then $HN$ is a subgroup of $G\text{,}$ $H \cap N$ is a normal subgroup of $H\text{,}$ and

\begin{equation*} H / H \cap N \cong HN /N\text{.} \end{equation*}

Proof.

We will first show that $HN = \{ hn : h \in H, n \in N \}$ is a subgroup of $G\text{.}$ Suppose that $h_1 n_1, h_2 n_2 \in HN\text{.}$ Since $N$ is normal, $(h_2)^{-1} n_1 h_2 \in N\text{.}$ So

\begin{equation*} (h_1 n_1)(h_2 n_2) = h_1 h_2 ( (h_2)^{-1} n_1 h_2 )n_2 \end{equation*}

is in $HN\text{.}$ The inverse of $hn \in HN$ is in $HN$ since

\begin{equation*} ( hn )^{-1} = n^{-1 } h^{-1} = h^{-1} (h n^{-1} h^{-1} )\text{.} \end{equation*}

Next, we prove that $H \cap N$ is normal in $H\text{.}$ Let $h \in H$ and $n \in H \cap N\text{.}$ Then $h^{-1} n h \in H$ since each element is in $H\text{.}$ Also, $h^{-1} n h \in N$ since $N$ is normal in $G\text{;}$ therefore, $h^{-1} n h \in H \cap N\text{.}$

Now define a map $\phi$ from $H$ to $HN / N$ by $h \mapsto h N\text{.}$ The map $\phi$ is onto, since any coset $h n N = h N$ is the image of $h$ in $H\text{.}$ We also know that $\phi$ is a homomorphism because

\begin{equation*} \phi( h h') = h h' N = h N h' N = \phi( h ) \phi( h')\text{.} \end{equation*}

By the First Isomorphism Theorem, the image of $\phi$ is isomorphic to $H / \ker \phi\text{;}$ that is,

\begin{equation*} HN/N = \phi(H) \cong H / \ker \phi\text{.} \end{equation*}

Since

\begin{equation*} \ker \phi = \{ h \in H : h \in N \} = H \cap N\text{,} \end{equation*}

$HN/N = \phi(H) \cong H / H \cap N\text{.}$

Theorem 4.25. Correspondence Theorem.

Let $N$ be a normal subgroup of a group $G\text{.}$ Then $H \mapsto H/N$ is a one-to-one correspondence between the set of subgroups $H$ containing $N$ and the set of subgroups of $G/N\text{.}$ Furthermore, the normal subgroups of $G$ containing $N$ correspond to normal subgroups of $G/N\text{.}$

Proof.

Let $H$ be a subgroup of $G$ containing $N\text{.}$ Since $N$ is normal in $H\text{,}$ $H/N$ makes is a factor group. Let $aN$ and $bN$ be elements of $H/N\text{.}$ Then $(aN)( b^{-1} N )= ab^{-1}N \in H/N\text{;}$ hence, $H/N$ is a subgroup of $G/N\text{.}$

Let $S$ be a subgroup of $G/N\text{.}$ This subgroup is a set of cosets of $N\text{.}$ If $H= \{ g \in G : gN \in S \}\text{,}$ then for $h_1, h_2 \in H\text{,}$ we have that $(h_1 N)( h_2 N )= h_1 h_2 N \in S$ and $h_1^{-1} N \in S\text{.}$ Therefore, $H$ must be a subgroup of $G\text{.}$ Clearly, $H$ contains $N\text{.}$ Therefore, $S = H / N\text{.}$ Consequently, the map $H \mapsto H/N$ is onto.

Suppose that $H_1$ and $H_2$ are subgroups of $G$ containing $N$ such that $H_1/N = H_2/N\text{.}$ If $h_1 \in H_1\text{,}$ then $h_1 N \in H_1/N\text{.}$ Hence, $h_1 N = h_2 N \subset H_2$ for some $h_2$ in $H_2\text{.}$ However, since $N$ is contained in $H_2\text{,}$ we know that $h_1 \in H_2$ or $H_1 \subset H_2\text{.}$ Similarly, $H_2 \subset H_1\text{.}$ Since $H_1 = H_2\text{,}$ the map $H \mapsto H/N$ is one-to-one.

Suppose that $H$ is normal in $G$ and $N$ is a subgroup of $H\text{.}$ Then it is easy to verify that the map $G/N \rightarrow G/H$ defined by $gN \mapsto gH$ is a homomorphism. The kernel of this homomorphism is $H/N\text{,}$ which proves that $H/N$ is normal in $G/N\text{.}$

Conversely, suppose that $H/N$ is normal in $G/N\text{.}$ The homomorphism given by

\begin{equation*} G \rightarrow G/N \rightarrow \frac{G/N}{H/N} \end{equation*}

has kernel $H\text{.}$ Hence, $H$ must be normal in $G\text{.}$

Notice that in the course of the proof of Theorem 4.25 , we have also proved the following theorem.

Theorem 4.26. Third Isomorphism Theorem.

Let $G$ be a group and $N$ and $H$ be normal subgroups of $G$ with $N \subset H\text{.}$ Then

\begin{equation*} G/H \cong \frac{G/N}{H/N}\text{.} \end{equation*}

Example 4.27.

By the Third Isomorphism Theorem,

\begin{equation*} {\mathbb Z} / m {\mathbb Z} \cong ({\mathbb Z}/ mn {\mathbb Z})/ (m {\mathbb Z}/ mn {\mathbb Z})\text{.} \end{equation*}

Since $| {\mathbb Z} / mn {\mathbb Z} | = mn$ and $|{\mathbb Z} / m{\mathbb Z}| = m\text{,}$ we have $| m {\mathbb Z} / mn {\mathbb Z}| = n\text{.}$

Reading Questions Reading Questions

1.

If $\phi:G \to H$ is a group homomorphism, what can you say about $\ker \phi\text{?}$

2.

If $N$ is a normal subgroup of $G\text{,}$ must there be a homomorphism that has $N$ as its kernel? If so, what is it?

3.

What is the point of showing that $\eta(g_1Kg_2) = \eta(g_1K)\eta(g_2K)$ in the proof of Theorem 4.22 ? Also, why is the proof not done directly after showing this?

4.

After reading the section, what questions do you still have? Write at least one well formulated question (even if you think you understand everything).

Exercises Practice Problems

1.

Describe all the homomorphisms from $\Z_{24}$ to $\Z_{18}\text{.}$

Hint.

The kernel of a homomorphism must be a subgroup of the domain and the image must be a subgroup of the codomain. Further, a generator must map to a generator.

2.

Prove that $\Z/\langle 5\rangle \cong \Z_5\text{.}$

3.

Let $\FR$ be the set of all functions $f:\R to \R$ under addition. Let $B = \{f \in \FR \st f(0) = 0\}\text{.}$ Prove that $\FR/B \cong \R\text{.}$

Hint.

Define a surjective homomorphism $\psi:\FR \to \R$ and use the FHT.

4.

How many homomorphic images are there of $D_4\text{?}$

Exercises Collected Homework

C1.

Let $G$ be an abelian group. Let $H = \{g^2 \st g \in G\}$ and $K = \{g \in G \st g^2 = e\}\text{.}$

Find $H$ and $K$ for the example $G = U(7)\text{.}$
Use the Fundamental Homomorphism Theorem the relate the groups $H$ and $K\text{.}$ That is, find a surjective homomorphism from $G$ to one of them, such that the other is the kernel. Justify your claims (including that your function really is a homomorphism). Then explain how one is isomorphic to a quotient group involving the other.