Proving something is a normal subgroup?

Question

I know that $$H=\left<\begin{bmatrix}i&0\\0&-i\end{bmatrix},\begin{bmatrix}0&i\\i&0\end{bmatrix}\right>.$$

and $$N=\left <\begin{bmatrix}-1&0\\0&-1\end{bmatrix}\right >.$$

i want to prove that $N\unlhd H $

could anyone provide me with a method to approaching something like this?

Answer 1

The generator of $N$ commutes with everything; therefore $N$ is normal in each of its supergroups.

Answer 2

The generator of $N$ commutes with the generators of $H$; thus for all $n\in N, h\in H,$ $hnh^{-1}=n$, so $N$ is normal in $H$.