= Quaternion group modulo its centre
{title2=$Q_8/Z(Q_8)\cong C_2\times C_2$}
In the <quaternion group>, $-1=i^2=j^2=k^2$ commutes with each generator because any element commutes with its powers. Hence $N=\{1,-1\}$ is central and is a <normal subgroup>. The reverse products satisfy $ji=-k$, $kj=-i$, $ik=-j$, so none of $\pm i,\pm j,\pm k$ is central. Therefore $Z(Q_8)=N$.
The four <cosets> are $N,iN,jN,kN$. Each nonidentity coset squares to $N$, and $(iN)(jN)=kN$. Thus the <quotient group> is the <Klein four-group>. Explicitly, mapping $\pm1$ to its identity and mapping $\pm i,\pm j,\pm k$ to its three different nonidentity elements is a surjective <group homomorphism> with <group kernel> $N$. The <first isomorphism theorem> gives the title result.
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