Consider the following subsets of the group of $2\times 2$ non-singular matrices over $\mathbb{R}$:
$G=\left\{ \begin{bmatrix}a&b\\ 0&d\end{bmatrix}: a,b,d\in \mathbb{R}, ad=1\right\}$$H=\left\{ \begin{bmatrix}1&b\\ 0&1\end{bmatrix}: b\in \mathbb{R}\right\}$
Which of the following statements are correct?
- $G$ forms a group under matrix multiplication.
- $H$ is a normal subgroup of $G$.
- The quotient group $G/H$ is well defined and is abelian.
- The quotient group $G/H$ is well defined and is isomorphic to the group of 2×2 diagonal matrices (over reals) with determinant 1.
So I checked that option 1 is true. Also for all $g\in G$, $g^{-1}Hg=H$ so $H$ is a normal subgroup of $G$. Option 2 is also correct. But I am stuck with option 3 and 4. Since $H$ is normal, we can talk about the quotient group $G/H$, but I don't know how this quotient group is look like ? and also the well define part. Also I need help with the 4th option. How can I do this? Any help would be great. thanks.
Best Answer
Hint:
Let $K$ be the group of $2\times 2$ diagonal matrices (over reals) with determinant $1$.
Define $f:G\rightarrow K$ by $$\begin{pmatrix}a&b\\0&d\end{pmatrix}\mapsto \begin{pmatrix}a&0\\0&d\end{pmatrix}$$ Try to verify that this is a surjective homomorphism with kernel $H$.
Then apply the first isomorphism theorem.