Quaternions group $\{\pm 1 , \pm i, \pm j, \pm k\}$ is not isomorphism to Diedral Group $D_4$.

Question

How to prove that quaternions group $G=\{\pm 1 , \pm i, \pm j, \pm k\}$ is not isomorphism to Diedral Group $D_4$?

Answer 1

Just write the elements of both groups. For example in $Q_8$ we have only one element of order $2$ while $D_8$ has $5$ elements of order $2$.

Answer 2

You could work out their respective Cayley graphs and notice they are different.