Elements of Group Theory

In this appendix, we briefly review some of those concepts in group theory that are used in our discussion of symmetry. Most of our readers will no doubt be familiar with these concepts already. We therefore provide no proofs or examples these can be found in one of the many good introductory accounts of group theory (Atkins and Friedman 1997, Schonland 1965, Hamermesh 1962, Tinkham 1964). 

The elements of a group do not necessarily commute under the group multiplication, that is, we may in general have gh hg where g, e G. If all the elements of the group commute under group multiplication, then the group is called Abelian. 

The order of the group is the number of elements in the set. This need not be finite, and the set need not even be countable. 

A subgroup of G is a subset of G that itself fulfills the group requirements. G and e are considered improper subgroups of G, all other subgroups are proper. 

A homomorphism is a mapping from one group into another that preserves products. If p. denotes the mapping from G to G, such that gi,g2 e G have the images p(gi), p(g2) e G, then homomorphism requires that p gig2) = ii gi)p g2)- If every element of G is the image of an element of G, the mapping is onto the set G. A homomorphism that is 1-to-l and onto is called an isomorphism. 

---

A.W. Joshi, Elements of Group Theory for Physicists,M 1973, Halsted Press, N.Y.M... 

The direct comparison of adsorbate modes with the modes for the free ions in solution, without considering symmetry changes in the adsorbed state, is not the correct procedure for band assignment. One must use the elements of group theory with the aid of the surface selection rule in order to make a proper spectral analysis. 

---