I have $Q_8=\{I,A,A^2,A^3,B,AB,A^2B,A^3B\}$ where $A=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ and $B=\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$
Now I take a look at $Q_8/N$ for all $N\unlhd Q_8$ (i.e for which G is there an isomorphism $Q_8/N\cong G$ ??)
I know that $|N|\in\{2,4\}$. If $|N|=2$ then $N=\{I,-I\}$ and clearly $Q_8/N\cong V$ where V is Klein-Four group.
What if $|N|=4$ ?