Prove that $\mathbb {D}_4 $ is isomorphic with $\mathbb {Z}_2 \times\mathbb{Z}_2 \times\mathbb{Z}_2$

Question

I tried comparing characteristics between the Dihedral group and the $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ group, but I need help to be able to define the function and be able to carry out the corresponding demonstration. Thank you

Answer 1

They're not isomorphic.

The dihedral group $D_{4}$ of eight elements has an element of order four; in particular, the rotation of the square by $\frac{\pi}{2}$ about the origin will suffice. No element of $\Bbb Z_2\times \Bbb Z_2\times \Bbb Z_2$ has order four, since, as can be computed readily, each nontrivial element has order two. Isomorphisms preserve orders of elements.

Answer 2

$D_8$ is not abelian, but $\Bbb Z_2\times\Bbb Z_2\times\Bbb Z_2$ is.