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.

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.