# Isomorphism - Group Theory

• April 26th 2008, 02:14 PM
Alborg
Isomorphism - Group Theory
Are S8, Z8 and Dih(8) isomorphic to each other and why?
• April 26th 2008, 02:39 PM
TheEmptySet
Quote:

Originally Posted by Alborg
Are S8, Z8 and Dih(8) isomorphic to each other and why?

Since $\mathbb{Z}_8$ is a communitive group and $S_8$ (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.
• April 26th 2008, 02:43 PM
icemanfan
Quote:

Originally Posted by TheEmptySet
Since $\mathbb{Z}_8$ is a communitive group and $S_8$ (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.
• April 26th 2008, 02:47 PM
TheEmptySet
Quote:

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.
• April 27th 2008, 07:10 PM
ThePerfectHacker
Quote:

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 $D_{2n}$ for the dihedral group because the dihedral group on $n$ verticies is of order $2n$.