I need to find the order of the group generated by the matrices
$$\begin{pmatrix}0&1\\-1&0\end{pmatrix},\begin{pmatrix}0&i\\-i&0\end{pmatrix}$$
under multiplication.
$\begin{pmatrix}0&1\\-1&0\end{pmatrix}\begin{pmatrix}0&i\\-i&0\end{pmatrix}=\begin{pmatrix}-i&0\\0&-i\end{pmatrix}$
and $\begin{pmatrix}-i&0\\0&-i\end{pmatrix}^4=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ so $4$ is the order? am I right?
Best Answer
No. This group contains, e.g., $\begin{pmatrix}0&1\\-1&0\end{pmatrix}^2=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$.
Addendum: Let $A=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ and $B=\begin{pmatrix}0&i\\-i&0\end{pmatrix}$. We have $A^4=I, B^2=I, AB=BA$. Hence the group generated by them is $\langle A\rangle \times \langle B \rangle $ and has order 8.