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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).