An Elementary Introduction to Group Theory

A mathematical group is a collection of elements with a single method for combining two elements of the group. We call the method multiplication in order to exploit the similarities of this operation with matrix and operator multiplication. The following requirements must be met  

The group must contain the identity element, E, such that 

The inverse of every element of the group must be a member of the group. 

It is not necessary that the elements of the group commute with each other. That is, it is possible that 

If all the members of the group commute, the group is called abelian  

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DeKock, R. L., and Gray, H. B. (1980). Chemical Bonding and Structure. Benjamin Cummings, Menlo Park, CA. An excellent introduction to bonding that makes use of group theory at an elementary level. 

It is well known from structural and kinetic studies that enzymes have well-defined binding sites for their substrates (3), sometimes form covalent intermediates, and generally involve acidic, basic and nucleophilic groups. Many of the concepts in catalysis are based on transition state (TS) theory. The first quantitative formulation of that theory was extensively used in the work of H. Eyring (4, 5 ). Noteworthy contributions to the basic theory were made by others (see (6) for review). As an elementary introduction, we will apply the fundamental assumptions of the TS theory in simple enzyme catalysis as follows. 

Flurry, R. L. Jr., Symmetry Groups, Prentice-Hall, Englewood Cliffs, New Jersey, 1980. An excellent introduction to chemical applications of group theory. Our text assumes famil-arity with only the most elementary group theoretical ideas and notations. 

Sala was also an important champion of the introduction of the chemical medicines. Sala s description of fermentation, as an intimate movement of elementary particles which tend to group themselves in a different order to make new compounds, is evidence of a concept doubtless derived from the atomic theory of the Greeks, and differs from the concept of chemical action in the nineteenth century mainly by lacking qualitative and quantitative definition. 

The study of molecular vibrations will be introduced by a consideration of the elementary dynamical principles applying to the treatment of small vibrations. In order that attention may be focused on the dynamical principles rather than on the technique of their application, this chapter vill employ only relativelj familiar and straightforward mathematical methods, and the illustrations will be simple. This will serve adequately as an introduction to the applications of quantum mechanics and group theory to the problem of molecular vibrations. Since, how-ever, these straightforward methods become cumbersome and impractical, even for simple molecules, equivalent but more powerful techniques u.sing matrix and vector notations will be discussed in Chap. 4. 

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