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Section10.4The 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*}
by

\begin{equation*}
\phi(g) = gH.
\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 ).
\end{equation*}
The kernel of this homomorphism is \(H\text{.}\) The following theorems describe the relationships between group homomorphisms, normal subgroups, and factor groups.

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Theorem10.10First 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 \(\psi = \eta \phi\text{.}\)

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).
\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, but

\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 ).
\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 \(\psi = \eta \phi\text{.}\)

\begin{tikzpicture}[scale=0.8] \node at (1.5,2) [above] {$\psi$}; \node at (0.25,0.65) {$\phi$}; \node at (2.75,0.65) {$\eta$}; \draw [->] (0,2) node [left] {$G$} -- (3,2) node [right] {$H$}; \node at (1.5,0) [below] {$G/K$}; \draw [->] (0,1.7) -- (1.3,0); \draw [->] (1.7,0) -- (3,1.7); \end{tikzpicture}

######
Example10.11

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).
\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 \({\mathbb Z}_n\text{.}\)

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Theorem10.12Second 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.
\end{equation*}
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} ).
\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').
\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.
\end{equation*}
Since

\begin{equation*}
\ker \phi = \{ h \in H : h \in N \} = H \cap N,
\end{equation*}
\(HN/N = \phi(H) \cong H / H \cap N\text{.}\)

######
Theorem10.13Correspondence 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 \(G/N\text{.}\)

Let \(H\) be a subgroup of \(G\) containing \(N\text{.}\) Since \(N\) is normal in \(H\text{,}\) \(H/N\) makes sense. 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 10.13, we have also proved the following theorem.

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Theorem10.14Third 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}.
\end{equation*}
######
Example10.15

By the Third Isomorphism Theorem,

\begin{equation*}
{\mathbb Z} / m {\mathbb Z} \cong ({\mathbb Z}/ mn {\mathbb Z})/ (m {\mathbb Z}/ mn {\mathbb Z}).
\end{equation*}
Since \(| {\mathbb Z} / mn {\mathbb Z} | = mn\) and \(|{\mathbb Z} / m{\mathbb Z}| = m\text{,}\) we have \(| m {\mathbb Z} / mn {\mathbb Z}| = n\text{.}\)