Chemical Applications of Group Theory by F. Albert Cotton

By F. Albert Cotton

Keeps the easy-to-read structure and casual taste of the former versions, and contains new fabric at the symmetric homes of prolonged arrays (crystals), projection operators, LCAO molecular orbitals, and electron counting ideas. additionally includes many new workouts and illustrations.

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30) Prove that the group of all positive numbers under multiplication is isomorphic to tho group of all real numbers under addition. 31) Let G denote a cyclic group of order 12 generated by an element A and lot H be a subgroup generated by the element A3. Find all the cosets of H in G and obtain the multiplication table for the factor group G/H. (I. 32) Consider the set of the following six functions: /1 (x)=x, II (x)=l-x, 13 (x)=x/(x-l). I, (x)=l/x, 15 (x)=l/

Vi) n=6. There are again two distinct (nonisomorphic) groups. We shall prove only a part of this statement to illustrate the argument involved. Let us denote the group by (E, A, B, C, D, F) . As before, we note that the orders of all the elements except E must be 2, 3 or 6. If the order of anyone elements is 6, it follow that we have a cyclic group of order 6, (A, A2, AS, A', AS, A6=E). Therefore, to find the second possible structure we exclude this case. Now we shall show that not all the elements A, B, C, D and F can be of order 2.

Any group of order 4 must be isomorphic to one of these two groups. (v) n=5. Only one distinct structure is possible in this case: the cyclic group of order 5, (A, At, A3, A', AS=E). (vi) n=6. There are again two distinct (nonisomorphic) groups. We shall prove only a part of this statement to illustrate the argument involved. Let us denote the group by (E, A, B, C, D, F) . As before, we note that the orders of all the elements except E must be 2, 3 or 6. If the order of anyone elements is 6, it follow that we have a cyclic group of order 6, (A, A2, AS, A', AS, A6=E).

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