Showing these two groups are both non-isomorphic, non abelian and have order 12

Question

I am trying to show that the groups of rotations of a regular tetrahedron and a regular hexagonal prism are not isomorphic, not abelian and have order 12.

So the set of rotations of the regular tetrahedrons has 12 elements, so that has order 12. I am unsure how to show it is non abelian however. As for the hexagonal prism, I am familiar with the rotations of a regular hexagon, but not a hexagonal prism and have no idea what the group of rotations would be.

P.S. A similiar small question I had is that is the group of rotations of a regular tetrahedron isomorphic to $\mathbb{Z}_{12}$? They are both of order 12. However, the tetrahedron group is apparently not abelian so that would mean they are not isomorphic, correct? Thanks!

Answer 1

The hexagonal prism has a rotation with period $6$. Does the tetrahedron?

Answer 2

Hints:

• The hexagonal prism has a rotation of order $6$. The tetrahedron not. Remember that an isomorphism maps element of a certain order to an element with the same order.
• To show a group is non-abelian, it suffices to find $2$ elements that not commute. Think about this geometrically. When I apply one transformation, and then another, does it yield the same result when I reverse the order? Translate back to group theory.
• $\mathbb{Z}_{12}$ is cyclic, and therefore abelian. So, if both of your groups are non abelian, they can't be isomorphic to this group.