Let G be the group of all nonsingular lower-triangular 2x2 matrixes (with non-zero determinant) with $\mathbb{R}$ coefs. Proof that all matrixes that are contained in $G$ and look like this: $\begin{pmatrix} a & 0 \\ * & a^2 \end{pmatrix}$ form a normal subgroup in G.
P.S I was trying to apply following lemma: for $H$ being subgroup of $G$ than $H$ may be called normal subgroup if and only if $gHg^{-1} \subseteq H\ \forall g \in G $. However it is hard form me to apply it here. Any solutions?
On
You must show that for $$\begin{bmatrix}u&0\\v&w\end{bmatrix}\in G,\begin{bmatrix}a&0\\b&a^2\end{bmatrix}\in H$$ there exists $$\begin{bmatrix}x&0\\y&z\end{bmatrix}\in H$$such that$$\begin{bmatrix}u&0\\v&w\end{bmatrix}\begin{bmatrix}a&0\\b&a^2\end{bmatrix}=\begin{bmatrix}x&0\\y&z\end{bmatrix}\begin{bmatrix}u&0\\v&w\end{bmatrix}$$Writing out the equations for row 1, column 1; row 2,column 1; row 2, column 2, we have $$ua=xu,$$ $$va+wb=yu+zv$$ $$wa^2=zw$$, so $$x=a,z=a^2,$$ $$y=\frac{va+wb-a^2v}{u}$$ so $$\begin{bmatrix}x&0\\y&z\end{bmatrix}=\begin{bmatrix}a&0\\\frac{va+wb-a^2v}{u}&a^2\end{bmatrix}\in H$$
Hint:
Let $A=\begin{pmatrix} a_{11} & 0 \\ a_{21} & a_{22} \end{pmatrix}$ be in given $G$ such that $\det A \neq 0$ and suppose $B=\begin{pmatrix} a & 0 \\ * & a^2 \end{pmatrix}$ is in the given subgroup $H$.
Now, try to show that $ABA^{-1}$ is of the form $\begin{pmatrix} b & 0 \\ c & b^2 \end{pmatrix}$, where all the coefficients are real.