Cosets

Theorem 2 The order of a subgroup of a finite group is a divisor of the order of the group. 

Consider a group G and subgroup H with respective orders G and. The theorem states that G / f is an integer. The proof is based on two elements. One first has to prove that all elements in a given coset are different and further that different cosets do not manifest any overlap. 

Consider a coset RiH with elements Rihx. For hx hy, Rihx must be different from Rihy, simply because two elements in the same row in the multiplication table can never be equal, as was proven in Eq. (3.9). Hence, the size of the coset will be equal to H. Then we consider an element Rj i RiH. This new element will in turn be the representative of a new coset, RjH, and we must prove that this new coset does not overlap with the previous one. This can easily be demonstrated by reductio ad absurdum. Suppose that there is an element Rjhx in the second coset that also belongs to the first coset, as Rihy. We then have  

Since the subgroup H has the group property, the inverse element is also an 

Rj e Ri H, contrary to the assumption. The expansion of the group in cosets thus leads to a complete partitioning in subsets of equal sizes. It starts by the subset formed by the subgroup H. If H is smaller than G, take an element outside H and form with this element a coset, which will have the same dimension as H.lf there are still elements outside, use one of these to form a new coset, again containing H  

---

RucE72 Ruch, E. Algebraic aspects of the chirality phenomenon in chemistry. Accounts of Chem. Res. 5 (1972) 49-56. RucE83 Ruch, E., Klein, D. J. Double cosets in chemistry and physics. Theor. Chim. Acta 63 (1983) 447-472. 

Case (a) a0, reproduces the set of functions The irreducible corepresentation D of 0 corresponds to a single irreducible representation A (u) of H, and has the same dimension. In this case no new degeneracy is introduced by the coset ua0. 

These five processes have been defined independently of the ligand partition. They are visualized as properties of the skeleton symmetry only. In the coset and double coset formulations of stereoisomerization this idea is expressed in a precise mathematical form. The underlying assumption is that the presence of different ligands does not distort the skeleton geometry. It is certainly possible to find chemical situations where this is reasonably correct. 

---