# Thread: Isomorphism - Group Theory

1. ## Isomorphism - Group Theory

Are S8, Z8 and Dih(8) isomorphic to each other and why?

2. Originally Posted by Alborg
Are S8, Z8 and Dih(8) isomorphic to each other and why?
Since $\displaystyle \mathbb{Z}_8$ is a communitive group and $\displaystyle S_8$ (Symmetric group of 8 elements)is not they cannot be isomophic.

You cannot preserve the group operation.

Sorry I don't know what Dih(8) stands for

Good luck.

3. Originally Posted by TheEmptySet
Since $\displaystyle \mathbb{Z}_8$ is a communitive group and $\displaystyle S_8$ (Symmetric group of 8 elements)is not they cannot be isomophic.

You cannot preserve the group operation.

Sorry I don't know what Dih(8) stands for

Good luck.
Dih(8) is the dihedral group of order 8, i.e. the symmetries of the octagon, with the actions of flipping and rotating 45 degrees.

4. Originally Posted by icemanfan
Dih(8) is the dihedral group of order 8, i.e. the symmetries of the octagon, with the actions of flipping and rotating 45 degrees.
Cool I just looked it up in a textbook and it has 16 elements. So it cannot be isomorphic to either of the above two groups because there is not a bijection(1-1 onto function) between the groups.

I hope this helps.

Good luck.

5. Originally Posted by TheEmptySet
Cool I just looked it up in a textbook and it has 16 elements.
It is conflicting notation. Some books use $\displaystyle D_{2n}$ for the dihedral group because the dihedral group on $\displaystyle n$ verticies is of order $\displaystyle 2n$.