Let $(T,*)$ be the subgroup of $\text{GL}_2(\mathbb{R})$ consisting of
nonsingular upper-triangular $2\times 2$ matrices with entries in
$\mathbb{R}$; that is the matrices of the form $$ \begin{pmatrix} a & b\\ 0 & c \end{pmatrix} $$ with $a,b,c$ elements of $\mathbb{R}$ and
$ac$ not equal to $0$. Let $U$ consist of matrices of the form
\begin{pmatrix} 1 &x\\ 0 &1 \end{pmatrix} with $x \in \mathbb{R}$.
Show that $U$ is a normal subgroup of $T$.
To show that $U$ is a normal subgroup of $T$, I multiplied the matrices, $UTU^{-1}$ and showed that $UTU^{-1}$ is a subset of $T$.
I believe my calculations are correct, but from my calculations it seems that $UTU^{-1}$ is not a subset of $T$.
I know there are other ways to show that $U$ is a normal subgroup of $T$. Any suggestions and corrections would be much appreciated!
Best Answer
A subgroup $H < G$ is normal if it's invariant by conjugation, so if for any $g \in G$ , $gHg^{-1} = H$.
Checking this is pretty easy:
Pick an arbitrary matrix in $T$, $\begin{pmatrix} a & b\\ 0 & c \end{pmatrix}$, compute its inverse $\begin{pmatrix} 1/a & -b/(ac)\\ 0 & 1/c \end{pmatrix}$ (note that, since the matrix is an element of the general linear group, it is invertible, so neither $a$ nor $c$ can be zero ) and verify that $$\begin{pmatrix} a & b\\ 0 & c \end{pmatrix} \begin{pmatrix} 1 & x\\ 0 & 1\end{pmatrix}\begin{pmatrix} 1/a & -b/(ac)\\ 0 & 1/c \end{pmatrix} = \begin{pmatrix} 1 &*\\ 0 & 1 \end{pmatrix}$$ (I'll leave the actual multiplication to you).