Is $O(2,\mathbb{R})$, the group of orthogonal $2\times2$ matrices a normal subgroup of $GL(2,\mathbb{R})$, the group of invertible $2\times2$ matrices?
My attempt: If $O(2,\mathbb{R})$ were a normal subgroup, then for any $A$ in $O(2,\mathbb{R})$ and any $G$ in $GL(2,\mathbb{R})$, we would have that $GAG^{-1}\in O(2,\mathbb{R})$, i.e. $(GAG^{-1})(GAG^{-1})^T=I$. This does not seem immediately apparent to me, so I'm trying to think of a simple counterexample. Any thoughts?
Any help appreciated!
Best Answer
$$ \begin{bmatrix} 1 & 0\\ 0 & -1\\ \end{bmatrix} $$ is orthogonal, but $$ \begin{bmatrix} 1 & 1\\ 0 & 1\\ \end{bmatrix}^{-1} \begin{bmatrix} 1 & 0\\ 0 & -1\\ \end{bmatrix} \begin{bmatrix} 1 & 1\\ 0 & 1\\ \end{bmatrix} = \begin{bmatrix} 1 & 2\\ 0 & -1\\ \end{bmatrix} $$ is not.