This software produces the following information about

Chapter 2, Problem 1CE

(choose chapter or problem)

This software produces the following information about U(n) (see Example 11).a. The elements of U(n).b. The inverse of each member of U(n).Run your program for n = 12, 15, 30, 36, 63.Exercise 2. This software determines the size of U(k). Run the program for k = 9, 27, 81, 243, 25, 125, 49, and 121. On the basis of this output try to guess a formula for the size of U(pn) as a function of the prime p and the integer n. Run the program for k = 18, 54, 162, 486, 50, 250, 98, and 242. Make a conjecture about the relationship between the size of U(2pn) and the size of U(pn) where p is a prime greater than 2.Exercise 3. This software computes the inverse of any element in GL(2,Zp), where p is a prime. Exercise 4. This software determines the number of elements in GL(2,Zp) and SL(2,Zp) where p is a prime. (The technical term for the number of elements in a group is the order of the group.) Run the program for p = 3, 5, 7, and 11. Do you see a relationship between the orders of GL(2,Zp) and SL(2,Zp) and p-1 ? Does this relationship hold for p=2 ? Based on these examples does it appear that p always divides the order of SL(2,Zp) ? What about p-1 ? What about p+1 ? Guess a formula for the order of SL(2,Zp). Guess a formula for GL(2,Zp).