Symmetric Elements

Desymmetrization, which refers to a process of efficiently desymmetrizing maw-molecules or achiral molecules to produce chiral ones, is a versatile method for preparing chiral nonracemic molecules.90 Desymmetrization of meso-compounds generally leads to the formation of a C-C or a C-X (X is a hetero atom) bond. The reaction normally uses a functional group residing on the symmetric element (in most cases the C2 axis or a plane) to differentiate two (or more) symmetrically equivalent functionalities elsewhere within the substrate molecule. This work was first reported by Hoye et al.91 and Mislow and Siegel92 in 1984. 

Lemma 1.2.1 If S has odd valency, all symmetric elements in S 1 have even valency. 

In the third section of this chapter, we investigate closed subsets of S generated by elements of valency 2. We start with closed subsets of S which are generated by a single symmetric element of valency 2. After that, we shall look at more general cases. 

Lemma 3.3.1 Let s be a symmetric element of S, and assume that s has valency 2. Then each element of (s) has valency at most 2. 

Let us now look at the structure of closed subsets of S generated by an arbitrary (not necessarily symmetric) element of valency 2. 

In this section, S is assumed to have finite valency. We shall first prove that a closed subset of S which is generated by a single symmetric element of valency 2 is faithfully embedded in S. After that we shall look at closed subsets in which each non-identity element has valency 2. 

The second theorem says that finite dihedral groups provide the only examples for schemes of finite valency which are generated by a single symmetric element. 

Synthesis of Microporous Compounds with Chiral Channels or Chiral Structural Features In theory, there are 66 chiral space groups among 230 space groups, such as PA, P43, P6i, P62, P3i, P3>2, P65, P64, etc. The framework structures crystallized in these space groups do not contain any symmetric element, and this results in the chiral structures. However, the occurrence of chiral frameworks is rare, and an optically pure chiral zeolite material has never been found. Among the large variety of zeolites and related compounds, only a few are known to have chiral channels or chiral structural features. Therefore, it is difficult to summarize the rules for the synthesis of such compounds. Herein, we will present a rough overview of the synthesis of chiral microporous structures or microporous compounds with chiral structural features based on their framework compositions. 

Moreover, from Lemma 1.1.2(iii), we obtain R = R. Thus, as A is regular, R contains no non-symmetric element. (Otherwise, A would have two equal columns.)... 

Theorem 9.5.3 Assume that S 1 contains symmetric elements and q such that p q and l,p, q = S. 

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