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 Zn. The software lists the elements of Zn
that generate all of Zn-that is, those elements k, 0
<= k <= n - 1, for which Zn = <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 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 .