Quaternion group

The quaternion group is $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ with $i^2=j^2=k^2=ijk=-1$. It is a nonabelian group of order eight: $ij=k$ but $ji=-k$.

Its unique element of order two is $-1$, and its other six nonidentity elements have order four. Every subgroup is normal: the three order-four subgroups have index two, and $\{1,-1\}$ is central. Thus it is a nonabelian Dedekind group. It differs from the dihedral group of order eight, which has several elements of order two, despite having the same group order.