Chapter 3

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 exercise repeats Exercise 1 for Z_n. The software lists the elements of Z_n that generate all of Z_n-that is, those elements k, 0 <= k <= n - 1, for which Z_n = <k>. 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 Z_n using a + b in place of ab.

Exercise 5. This software computes the order of elements in the GL(2,Z_p). 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 .

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