Q:

Prove normal of direct product of GL

Let $G=(\mathrm{GL}(2,\mathbb{R}) \oplus \mathrm{GL}(2,\mathbb{R}))$ and let $H = \{(A,B) \in G \mid\det(A)=\det(B)\}$. Prove that $H$ is normal in $G$.

Mostly confused on what $G$ is.

A:

$G$ is a collection of pairs of matrices, and $H$ contains those pairs that have the same determinant, with the operation defined by matrix multiplication component-wise.  For example,
$$\left(\begin{pmatrix}1&0\\0&1\end{pmatrix},\begin{pmatrix}1&1\\0&1\end{pmatrix}\right)$$
is an example of an element in $H$ because both matrices have determinant $1$.
Now, see if you can prove that $H$ is normal using the fact that $\det(ABA^{-1})=\det(B)$.

