Double coset

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. 

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. 

Such a procedure corresponding to the permutation-nomenclature system of configuration 7> is in accordance with the fact that a chemical constitution is a representative of a product D, X. .. X Da of the double cosets D, of pairs of subgroups symmetric groups Sn,g. The latter correspond to permutations of atoms of the same element and the symmetry of the adjacent subset of graph points whose degree is compatible with the coordination number of the element equivalence class of the atom. 

Ifpx attains its minimum, it does so on exactly one double coset G g/G... 

We have already referred to double cosets in Sec. 1.3 they are obtained as sets UGV, where G Q and U, V C Q. Each double coset is either distinct or equal to another they thus provide a disjoint partitioning of the elements of <7, termed a double coset decomposition [8]. Each element within a given double coset occurs Ag times, where... 

Unfortunately, this is not the end of the story, since the summation over H may contain redundant terms. To eliminate these, we use another double coset decomposition, based on the stabilizers A4 =Z/flV, which leaves the charge distribution faAfbR(B) invariant, and A/ = WDgeneral case.) Using a set of DCRs T obtained from M and Af, we can then replace our sum over H to give... 

Of these possibilities (as Table 17.17 shows) only the pair da = E,db=E are double coset representatives that satisfy (eq. 9),... 

The first section of this chapter is a collection of general observations on closed subsets most of which are straightforward generalizations of facts on subgroups. For instance, we show that the set of all double cosets of two closed subset of S is a partition of S cf. Lemma 2T.3. We also introduce transversals (as they have been introduced in [45]), and, at the end if this section, we show that closed subsets give rise to subschemes. 

Elements of this orbit are thus labeled by double labels (r, p). If we want this orbit to be doubly transitive the elements of G should be able to effectuate any mapping between pairs of such labels. It can easily be shown that this is not the case. Take for example the case where gr is the unit element e, and consider the initial pair (e, p), (e, ct). Both these cosets thus belong to H. Now we wonder if there would exist an element gx that maps this pair onto a pair (e, p ), (r, a ). In this target pair the first coset stays in H, but the second is outside H. 

In a separate paper, Fujita jointly with Sherif El-Basil57 used the concept of doubly-colored graphs to visualize the abstract concepts such as subductions of coset representations, double cosets, and unit-subduced-cycle-indices,58 which had been mathematically formulated in the framework of coset algebraic theory developed by Fujita.42... 

The enumeration of constitutional isomers by Cayley, then by Henze and Blair, and finally by means of Polya s theorem or double cosets was reviewed earlier. 

More recently, Ruch (a theoretical chemist formerly at West Berlin s Free University) developed a different proach based on group theory (double coset formalism) which can be successfully employed for isomer counting [48]. This approach entails less labour than the expansions of lengthy pdynomial required by Pdlya s theorem, but at the same time allows less flexibility in terms of figure wdghting. 

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