Exercise 4. This software determines the order of the group of units in the ring of \(2\) by \(2\) matrices over \(Z_n\) (that is, the group \(GL(2,Z_n)\) and the subgroup \(SL(2,Z_n)\). Run the program for \(n = 2, 3, 5, 7, 11\), and \(13\). What relationship do you see between the order of \(GL(2,Z_n)\) and the order of \(SL(2,Z_n)\) in these cases? Run the program for \(n = 16, 27, 25\), and \(49\). Make a conjecture about the relationship between the order of \(GL(2,Z_n)\) and the order of \(SL(2,Z_n)\) when \(n\) is a power of a prime. Run the program for \(n = 32\). (Notice that when you run the program for \(n = 32\) the table shows the orders for all divisors of 32 greater than 1.) How do the orders the two groups change each time you increase the power of \(2\) by \(1\)? Run the program for \(n = 27\). How do the orders the two groups change each time you increase the power of \(3\) by \(1\)? Run the program for \(n = 25\). How do the orders the two groups change when you increase the power of \(5\) by \(1\)? Make a conjecture about the relationship between \(|SL(2,Z_{p^i})|\) and \(|SL(2,Z_{p^{i+1}})|\). Make a conjecture about the relationship between \(|GL(2,Z_{p^i})|\) and \(|GL(2,Z_{p^{i+1}})|\). Run the program for \(n = 12, 15, 20, 21\), and \(30\). Make a conjecture about the order of \(GL(2,Z_n)\) in terms of the orders of \(GL(2,Z_s)\) and \(GL(2,Z_t)\) where \(n = st\) and \(s\) and \(t\) are relatively prime. (Notice that when you run the program for \(st\) the table shows the values for \(st\), \(s\) and \(t\).) For each value of n is the order of \(SL(2,Z_n)\) divisible by \(n\)? Is it divisible by \(n + 1\)? Is it divisible by \(n - 1\)?

Please enter \(n\) for \(Z_n\) and click the button, the order of \(GL(2,Z_n)\) and \(SL(2,Z_n)\) will show below.
\(n\):

Please enter \(s\) and \(t\) for \(Z_{st}\) and click the button, the order of \(GL(2,Z_{st})\) and \(SL(2,Z_{st})\) will show below.
\(s\): \(t\):