The Multiplication Table

It is noted that two successive symmetry transformations of a system leave that system invariant. The product of the two operations is therefore also a symmetry operation of the system. The set of symmetry transformations is therefore closed under the law of successive transformations. An identity transformation that leaves the system unchanged clearly belongs to the set. It is not difficult to see that any given symmetry transformation has an inverse that also belongs to the set. Since successive transformations of the set obey the associative law it finally follows that the set constitutes a 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. 

Groups with similar multiplication tables have the same structure and are said to be isomorphic. There exists an isomorphism between two groups G = E, A, B, C. and G = E, A, B, C. .. of the same order, if there is a one-to-one correspondence between the elements of G and G. A multiplication such as AB = C in one group then implies that A B = C in 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 

---

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). 

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

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. 

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. 

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). 

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. 

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. 

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... 

The left element on each line is a name for the field (or record) which is shown in the right element. The ORDER of this group is 2. The OPERAND list used for this example was the numbers 4 and 5. The Gensym symbols for the two group elements are stored in the GROUP-LIST field. As was explained earlier, property lists were attached to each of these which contained the multiplication table and other information. The elements of the CLASS-LIST and CHARACTER-LIST fields contain the information indicated by their names. In the above examples, we did not work with the subgroup of S2, so NIL is stored there but the direct product name S3-DP-S2 is stored in the field SUPERGROUPS. Attached to this name (not shown) is the association list for the correlation between the representations, which was used for the construction of Table IV. 

The covering operations of this molecule are only the identity E and a reflection in the plane of the molecule, denoted simply as a. Compare the multiplication table of this group with that of the group formed by 1 and -1 under multiplication ... 

Write the multiplication table for the group of permutations of three objects. 

Check out the multiplication table of the largest bottom number until you find a number that all the other bottom numbers evenly divide into. 

A useful theorem states that in a given row or a given column of the multiplication table of a group, each element appears only once thus each row and each column of the table is some permutation of the group elements. The proof is as follows Suppose the element F appeared twice in the row having B as the left member of the product. We would have... 

For groups of order 4, there are two possible forms for the multiplication table (Problem 9.12). 

Since there is only one possible structure for the multiplication table of a group of order 2, all groups of order 2 are isomorphic. Thus the point groups Gs> 6,-, and G2 are isomorphic also, the group of elements + 1 and — 1 with the rule of combination being ordinary multiplication is isomorphic to each of Gs, (2,., and G2. 

The multiplication table for the group G3v is given in Table 9.1. The three symmetry planes oa, ob, oc, are defined in Fig. 9.1 as making angles of 30°, 150°, and 270°, respectively, with the positive x axis. (The subscripts have no reference to the principal axes of inertia.) Consider the entry in the... 

Let us consider group multiplication tables in an abstract way, divorced from any particular group. Consider a group of order 1 what does its multiplication table look like The single element must be the identity element /, and the multiplication table has the single entry 11=I. 

Each row and each column contains each element exactly once (as must be true for the multiplication table of a group). Show that the above table is not the multiplication table of a group. 

For a group of order 3, the multiplication table will have to be, in part, as follows ... 

From this we find that the multiplication table, in the usual format, is as follows ... 

---