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.
$s$: $t$: