# I Group theory in physics

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1. Mar 27, 2017

### Konte

Hi everybody,

I have a question: is an abelian group can be isomorphic to a non-abelian group?

Thank you everybody.

2. Mar 27, 2017

### DrDu

In a non-abelian group, there are at least two elements A and B so that A*B=C but B*A=D with different D and C. this is not possible in an abelian group, so the two groups can also not be isomorphic.

3. Mar 27, 2017

### Konte

I thought so. But I asked this question because I have read this article of G.Bone and Co. http://dx.doi.org/10.1080/00268979100100021(in appendix page 71-72), which states that the Molecular symmetry group (MS) of a rigid molecule is isomorphic to its point group. Knowing that the MS group is composed of nuclear permutation-inversion operation, I verified by myself this group is in general non-abelian, and on the other side, point group are in general abelian. So, did they say something wrong?

Thank you.

Konte

4. Mar 27, 2017

### DrDu

Most point groups aren't abelian. Take a cubic group, like Oh as an example: Rotate first 90 deg, around x and then 90 deg. around y. Compare to first rotating 90 deg around y and then 90 deg around x.

5. Mar 27, 2017

### Konte

You are right, I missed it.
Thanks a lot.

I have another question, how to construct the character table of irreducible representation knowing the multiplication table of the group?

6. Mar 27, 2017

### DrDu

This can be arbitrarily nontrivial. Best have a look on a decent book on group theory. As you are mainly interested in the symmetric group, Sternberg, "Group theory and physics" might be what you are looking for.

7. Mar 27, 2017

Ok. Thanks!