**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*^{-1}*BA|* and * |B*^{-1}*AB|*.
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*^{-}^{1}*BA|* ? Do you see
any relationship between * |A| * and * |B*^{-}^{1}*AB|*
? Make a conjecture about this relationship. Test your conjecture for several
other choices for * A * and * B *.