Multiplication tables

This is not the same as (equation Al.4.14). In fact, in this convention, which we can call the S-convention, the multiplication table is the transpose of that given in table Al.4.1. The convention we use and which leads to the multiplication table given in table Al.4.1. will be called the N-convention (where N denotes nuclear-fixed labels). 

The complete multiplication table of the point group, worked out using arguments similar to those leading to (equation A1.4.23) and (equation A1.4.24), is given in table A1.4.2. It is left as an exercise for the reader to use this table to show that the elements of satisfy the group axioms given in section Al.4.2.1. 

Table Al.4.2 The multiplication table of the point group using the space-fixed axis convention (see text).

If we were to define the operations of the point group as also rotating and reflecting the (p.q.r) axis system (in which case the axes would be tied to the positions of the nuclei), we would obtain a different multiplication table. We could call this the nuclear-fixed axis convention. To implement this the protons in the o, O2 and planes in figure Al.4.2 would be numbered H, H2 and respectively. With this convention the operation would move the a plane to the position in space originally occupied by the 02 plane. If we follow such a C3 operation by the reflection (in the plane containing Ft ) we find that, in the nuclear-fixed axis convention ... 

As an example we consider the group introduced in (equation Al.4,19) and the point group given in (equation Al.4.22). Inspection shows that the multiplication table of in table Al.4,2 can be obtained from the multiplication table of the group (table Al.4,1) by the following mapping ... 

Thus, the matrices will have a multiplication table with the same structure as the multiplication table of the synnnetry group and hence will fonn an /-dimensional representation of the group. 

Rechen-schieber, -stab, m. slide rule, -stift, m. slate pencil, -tafel, /. reckoning table multiplication table nomograph, nomogram slate blackboard counting board, -ver-fahren, n. method or process of calculation. 

What s seven times nine Sixty-three, of course. You didn t have to stop and figure it out you knew the answer immediately because you long ago learned the multiplication tables. Learning the reactions of organic chemistry requires the same approach reactions have to be learned for immediate recall if they are to be useful. 

First, we recall the five different stereoisomerization processes these are necessary to reach any isomer from a given one in one step. However, they are not really independent because a succession of two processes is a linear combination of processes. A multiplication table of the processes has been established. This is explained in Sections II and III. 

Second, the symmetry properties of one of the processes (the Berry step) are analysed. The operators associated with it are shown to commute with the elements of a cyclic group of order ten. Because of the structure of the multiplication table, the same is true for the operators associated with the other stereoisomerization processes. The solution of the rate equations for any process are derived from these properties (Sections IV and V). 

The multiplication table for the processes has been given previously " and is repeated here for further reference (Table 1). It was... 

As a second example, consider the group formed by the elements 1, r, —1, -i, where i2 = -1. These elements are developed by the operations r , where n is an integer. If the law of combination is ordinary multiplication, the multiplication table for this cyclic group can be developed (Table 1). This... 

The ensemble of elements that are mutually conjugate form a class TJje concept of a class is most easily demonstrated by an example. The multiplication table for the group of matrices defined by Eq. (2) is given in Table 3. With its use the relations... 

Develop Ihble 3, the multiplication table for the matrices given by Eqs. (2). 

Table 2 Multiplication table for the four-group composed of the matrices of Eqs. (1). 

It is not too difficult to develop the multiplication table shown as Table 3 (problem 1). It will be noticed immediately that the table is not symmetric with respect to the principal diagonal Therefore, the group is not Abelian and multiplication is not commutative. 

Ttoo groups are isomorphic [Greek, too- (same or equal) +(iop rf (form)] if they have tbe same multiplication table. 

The four operations which form the symmetry group for the water molecule are represented in Fig. 2. It can be easily verified that the multiplication table for these symmetry operations is that already developed (Table 2). Thus, the symmetry group of the water molecule is isomorphic with the four-group. 

All of these combinations of operations can be summarized in a group multiplication table like that shown in Table 5.2. The multiplication table (see Table 5.2) for the C2v group is thus constructed so that the combination of operations follow the four rules presented at the beginning of this section. 

This process could be continued so that all the combinations of symmetry operations would be worked out. Table 5.3 shows the multiplication table for the C3 point group, which is the point group to which a pyramidal molecule such as NH3 belongs. 

Multiplication tables can be constructed for the combination of symmetry operations for other point groups. However, it is not the multiplication table as such which is of interest. The multiplication table for the C2v point group is shown in Table 5.2. If we replace E, C2, and cryz by +1, we find that the numbers still obey the multiplication table. For example,... 

Use the procedure outlined in the text to obtain the multiplication table for the C4i, point group. 

Because of the group property of closure all operations of any group can be presented in the form of a multiplication table that contains all elements of the group, e.g. 

It is noticed from the multiplication table that each element of the group occurs once and only once in each row or column. The arrangement of elements in a row (column) is different from that in any other row (column)1. An important consequence of this arrangement is that if / is any function of the group elements, then... 

Mathematically this group is known as a cyclic group of order n, generated by the element c, and consisting of the elements e, c, c2,. ..,cn-1. The multiplication table of C3 is... 

Using the rules of matrix multiplication it is further shown that these matrices reproduce the C2v multiplication table shown in Table 2 and readily obtained from the diagram below. As an example, from the table it follows that avC2 = cr v as shown by the matrix multiplication. 

Looked upon purely as an arithmetic multiplication table, the products are all correct. This condition remains valid whatever row is selected initially4. Because of the close association between tables 1 and 2 each row in Table 1 may indeed be regarded as a representation of the symmetry elements. 

A search for alternative sets that give a correct multiplication table in the same sense, is abortive. 

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