Left coset

These subgroups of G generated by multiplication of some element of G by the subgroup H are called cosets of H in G. If the multiplication is carried out on the left, they are called left cosets, and vice versa for right multiplication. 

A subgroup with the property given in Eq. 3.3 is referred to as the stabilizer of A in Q [4]. The transformed centres that can be generated from A may be found by applying operators chosen one from each distinct left coset of U in Q. The number of such cosets is the number of such transformed centres and is given by g/u, where g is the order of Q and u is the order of U. The quantity g/u is termed the index of the stabilizer, although the term constituency number has been used in some quantum chemical applications. 

In any formula involving a sum over group elements G, we can replace G with HG, where H is any group element, since the left coset of Q in Q is the group itself. Thus, for example,... 

The corresponding left coset expansion with the same coset representative is... 

Consequently, Hi is not an invariant subgroup. For H to be an invariant subgroup of G, right and left cosets must be equal for each coset representative in the expansion of G. 

The final step going from the small IRs of the little group G(k) to the IRs of G requires the theory of induced representations (Section 4.8). At a particular k in the representation domain, the left coset expansion of G on the little group G(k) is... 

For each thin scheme S, the group correspondence establishes a one-to-one correspondence between the closed subsets of S and the subgroups of S 7. This means, in particular, that the notion of a closed subset generalizes that of a subgroup. More importantly closed subsets retain some of the interesting properties of subgroups and transfer them to scheme theory. For instance, the set of all left cosets of a closed subset of a scheme S in S (defined with the help of the complex multiplication in S) is a partition of S. 4 Furthermore,... 

We briefly mention the case where the number of points m is less than n, i.e., less than the number of elements in the symmetry group G with respect to which we measure symmetry. In this case, m should be a factor of n such that there exists a subgroup H of G with n/m elements. In this case, we duplicate each point trim times and fold/unfold the points with elements of a left coset of G with respect to H. Following the folding/unfolding method, the relocated points will align on symmetry elements of G (on a reflection plane or on a rotation axis for example). Further details of this case and proof can be found in Ref. 2. 

Proof is immediate from the 1-1 relationship between points in the orbit of x and the left cosets of Gx. Each left coset of Gx consists of all elements of G that map x to a specific pointy). 

The model E is converted into the model (14) E by the ligand permutation (14). The action of (14) on the other models of X, as represented by the elements of Sx, yields the models of the left coset (14) Sx. The models of the isomer (14) X are obtained by the action of a ligand permutation (belonging to Sx, followed by the ligand permutation (14). These all belong to the permutation isomer (14) X, and they all correspond to this isomer... 

Correspondingly, in the theory of the chemical identity groups stereochemical features of the molecules and EMs are represented by their chemical identity groups and their left cosets in SymL, and the permutational isomerizations and the stereochemical aspect of chemical reactions are described by the so-called set-valued mappings of the left coset spaces of the respective chemical identity groups [10, 19, 35]. 

Let X and Y be isomers with the same set of ligands, and let the isomerization X - Y be a reference process, then processes with the same mechanism will convert X into those permutation isomers of Y that are represented by left cosets pSY with non-empty intersections X n pY 4= 0, and pY will also be analogously convertible into the permutation isomers XX, if we have XX n pY 4= 0. 

When chemical reactions with a constitutional aspect and a stereochemical aspect are written as ligand preserving reactions, then their stereochemical aspect is accounted for by set-valued mappings of the left cosets of the chemical identity groups of the EM of educts and the EM of products. Thus, for example, one can immediately tell how many and which stereoisomeric products may conceivably result from a given reaction [35],... 

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