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).