Chlorinated Isomers of Benzene

A more complicated example is to count the isomers of substituted benzene. Here the problem of symmetry becomes sound. We start now to elucidate the symmetries of the planar hexagon. 

In Fig. 12.2, these possibilities are listed. We bring now some order into the manifold. First, we label the comers of the box. We attach fixed numbers to the corners, just those which have the first benzene ring in Fig. 12.2 (left up). Then we know the possibilities, which are illustrated in Fig. 12.2 and also in Table 12.1. 

We rotate the ring, by pushing the first number to the rear. In this way, however, not all possibilities are exhausted. We can also mirror the ring. The symmetry oper- 

We put the cycle brackets to indicate that it is closed (135). The next higher number, which we did not go through yet, is the 2. We repeat the same procedure and obtain (246). Thus, we get with all positions from 1 to 6 for the special permutation 345612 (135)(246). In fact, we have subdivided the permutation 345612 into two subgroups of the same cycle length, i.e., 3. The whole procedure is a formal analysis, which usefulness we will recognize immediately. In the column length number, we have classified the permutations. If we finally arrange these permutations, then we obtain 

We must still standardize the generating function. If we set x equal to 0 and set h equal to 1, then a nonsubstituted isomer should be obtained with correct standardization. In this way, we receive the normalization factor. 

---

With the computation of the number of isomers of different chlorinated benzene molecules, a generating function arises that is connected with the symmetry of the benzene molecule. This function is designated after Pdlya as the Zyklenzeiger. The generating function for chlorine isomers of benzene is... 

---