Conjugate Elements and Classes

Exercise 1.1-3 Evaluate the products in the column headed P3 in Table 1.3. 

Exercise 1.1-4 (a) Using the multiplication table for S(3) in Table 1.3 show that (P3 P )P2 P3(Pi Pi - This is an example of the group property of associativity, (b) Find the inverse of P2 and also the inverse of P5. 

Exercise 1.1-1 (a) The set ] does not form a group because it does not contain the identity E. (b) The set p 0 contains the identity 0, + 0 p, but the inverses p of the elements p, p + (—p) = 0, are not members of the set p 0. (c) The set of positive and negative integers, including zero, p p 0, does form a group since it has the four group properties it satisfies closure, and associativity, it contains the identity (0), and each element p has an inverse p or —p. 

Exercise 1.2-1 Show that E is always in a class by itself. 

Example 1.2-1 Determine the classes of S(3). Note that P0=E is in a class by itself the class of E is always named P, . Using the multiplication table for S(3), we find 

---

In order to determine the classes within any particular group we can begin with one element and work out all of its transforms, using all the elements in the group, including itself, then take a second element, which is not one of those found to be conjugate to the first, and determine all its transforms, and so on until all elements in the group have been placed in one class or another. 

Designating a conjugate B to a symmetry operation A is also called a similarity transformation. B is a similarity transform of A by Z, or, in other words, A and B are conjugates. Elements belong to one class if they are conjugate to one another. The inverse operation can be applied with the aid of the multiplication table and rule 4 given above,... 

In (I) and (II) the sum is to be taken over symmetry operators R of different classes of conjugate elements only. — For xAR) different values have to be taken for non-linear (set a) and linear (set h) molecules. For linear molecules the characters for the operators C2 and a depend on whether the molecular axis is parallel,, or perpendicular, , to the symmetry elements. 

This means that A belongs to the same class together with B and C. Now, we will create some conjugate elements to D and F ... 

This procedure may seem quite complicated, but it is in fact very simple. Let us demonstrate this for the previous example, the set of the three U-orbitals on the hydrogens in NH3. We first determine the reducible character of this set (see x (I ) in Table 4.1). We do not have to do this for all six elements of 3 but only for one representative of each class since conjugate elements have the same characters. For the unit element, the representation matrix is of course the 3 x 3 unit matrix, and its trace is equal to three, the dimension of the set. For the other elements, we do not need to know the full matrix representation indeed, we need only the elements on the diagonal. Now a diagonal entry in a representation matrix can differ from zero only if a component function is turned into itself, or at least into a fraction of itself. 

Remark (The conjugacy classes of symmetric groups) We recall that two elements 7T and a of S are conjugate if and only if they have the same cycle partition, or, in other words, if and only if they are of the same cycle type ... 

Considering that the numbers and % are constant on the classes of conjugate subgroups, and that we would rather enumerate orbits by stabilizer class than individual elements by stabilizer, the number of equations can be substantially reduced as follows. LetH,K,... denote classes of conjugate subgroups of G, and... 

Define the following terms symmetry operation, symmetry element, principal axis, identity operation, improper rotation, inversion, symmetry group, point group, conjugate elements, similarity transformation and class. 

---