Solution: Exercise 1.This software determines the cyclic

Chapter 3, Problem 4CE

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Exercise 1.This software determines the cyclic subgroups of U(n) (n < 100). Run the program for n = 12, 15, and 30. Compare the order of the subgroups with the order of the group itself. What arithmetic relationship do these integers have ?Exercise 2. This software lists the elements of Zn that generate all of Zn-that is, those elements k, 0 <= k <= n - 1, for which Zn = . How does this set compare with U(n) ? (See computer exercise 1a in Chapter 2.) Make a conjecture.Exercise 3. This software does the following. For each pair of elements a and b from U(n), print out |a|, |b|, and |ab| on the same line. Assume n < 100. Run your program for n = 15, 30, and 42. What is the arithmetic relationship between |ab| and |a| and |b| ?Exercise 4. This exercise repeats exercise 3 for Zn using a + b in place of ab.Exercise 5. This software computes the order of elements in the GL(2,Zp). Enter several choices for matrices A and B. The software returns |A|, |B|, |AB|, |BA|, |A-1BA| and |B-1AB|. Do you see any relationship between |A|, |B| and |AB| ? Do you see any relationship between |AB| and |BA| ? Make a conjecture about this relationship. Test your conjecture for several other choices for A and B . Do you see any relationship between |B| and |A-1BA| ? Do you see any relationship between |A| and |B-1AB| ? Make a conjecture about this relationship. Test your conjecture for several other choices for A and B .