Let \(G\) be the group of real matrices \[ \begin{pmatrix} a&b\\ 0&a^{-1} \end{pmatrix}, \qquad a>0, \] and let \(N\) be the subgroup with \(a=1\).
Prove that \(N\triangleleft G\) and that \(G/N\cong(\mathbb R,+)\).
Find a proper normal subgroup of \(G\) which properly contains \(N\).
Solution
For \[ M= \begin{pmatrix} a&b\\ 0&a^{-1} \end{pmatrix} \in G, \] define \[ \phi(M)\coloneqq\log a. \]
1. The map\[ \phi:G\longrightarrow(\RR,+) \]is a group homomorphism.
If \[ M_1= \begin{pmatrix} a&b\\ 0&a^{-1} \end{pmatrix}, \qquad M_2= \begin{pmatrix} c&d\\ 0&c^{-1} \end{pmatrix}, \] then the upper-left entry of \(M_1M_2\) is \(ac\). Hence \[ \phi(M_1M_2) = \log(ac) = \log a+\log c = \phi(M_1)+\phi(M_2). \]
2. The homomorphism \(\phi\) is surjective and\[ \ker\phi=N. \]
For every \(t\in\RR\), the diagonal matrix \[ \begin{pmatrix} e^t&0\\ 0&e^{-t} \end{pmatrix} \] lies in \(G\) and maps to \(t\), so \(\phi\) is surjective.
Also, \[ \phi(M)=0 \iff \log a=0 \iff a=1, \] which is exactly the defining condition for membership in \(N\).
3. The subgroup \(N\) is normal in \(G\) and\[ \boxed{G/N\cong(\RR,+)}. \]
By step 2, \(N=\ker\phi\), so \(N\triangleleft G\). The first isomorphism theorem and surjectivity of \(\phi\) give \[ G/N \cong \operatorname{im}\phi = (\RR,+). \]
4. Define\[ H\coloneqq\phi^{-1}(\ZZ). \]Then \(H\) is a normal subgroup of \(G\).
The subgroup \(\ZZ\) is normal in the abelian group \((\RR,+)\). The inverse image of a normal subgroup under a group homomorphism is normal. Thus \(H\triangleleft G\).
5. The inclusions\[ N\subsetneq H\subsetneq G \]are both strict.
Since \(0\in\ZZ\), step 2 gives \[ N=\ker\phi\subseteq H. \] The matrix \[ \begin{pmatrix} e&0\\ 0&e^{-1} \end{pmatrix} \] lies in \(H\) because its image is \(1\), but it does not lie in \(N\), so the first inclusion is strict.
The matrix \[ \begin{pmatrix} e^{1/2}&0\\ 0&e^{-1/2} \end{pmatrix} \] lies in \(G\) but not in \(H\), because its image is \(1/2\notin\ZZ\). Hence the second inclusion is strict.
6. Therefore a proper normal subgroup properly containing \(N\) is\[ \boxed{H=\phi^{-1}(\ZZ)}. \]
Steps <1>4–<1>5 establish normality and both strict inclusions.
7. Q.E.D.
Step 3 proves part (a), and step 6 proves part (b).