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Problem 74 (Conjecturing The Order Of General Linear Groups)

Let's focus on 2 by 2 matrices with entries in $\mathbb{Z}_p$ where $p$ is a prime. So we let $p$ be a prime and consider $G = \text{GL}(2,\mathbb{Z}_p)$.

  1. Here is a list of all 81 of the 2 by 2 matrices with entries in $\mathbb{Z}_3$. Underneath that list you'll see the determinant of each matrix has already been computed for you. How many matrices have determinant 1, in other words how many matrices are in $\text{SL}(2,\mathbb{Z}_3)$. How many matrices are invertible, i.e. how many matrices are in $\text{GL}(2,\mathbb{Z}_3)$? What patterns did you use to help you count? 
    • n=3
matrices=[];
for i in [0..n-1]:
 for j in [0..n-1]:
  tempmatrices=[]
  for k in [0..n-1]:
   for l in [0..n-1]:
    tempmatrices.append(matrix(2,2,[i,j,k,l]))
  matrices.append(tempmatrices)
show(table(matrices))

dets=[];
for i in [0..n-1]:
 for j in [0..n-1]:
  tempdets=[]
  for k in [0..n-1]:
   for l in [0..n-1]:
    tempdets.append(matrix(2,2,[i,j,k,l]).det().mod(n))
  dets.append(tempdets)
show(table(dets))
  2. This sage code below repeats the above with $n=5$. Use this to count how many matrices have determinant 1 (are in $\text{SL}(2,\mathbb{Z}_5)$) and then how many are invertible (are in $\text{GL}(2,\mathbb{Z}_5)$). What patterns did you use to help you count? 
    • n=5  #Change this number as needed. 
matrices=[];
for i in [0..n-1]:
 for j in [0..n-1]:
  tempmatrices=[]
  for k in [0..n-1]:
   for l in [0..n-1]:
    tempmatrices.append(matrix(2,2,[i,j,k,l]))
  matrices.append(tempmatrices)
#show(table(matrices))  #Uncomment this line (remove the #) if you want to see all the matrices. 

dets=[];
for i in [0..n-1]:
 for j in [0..n-1]:
  tempdets=[]
  for k in [0..n-1]:
   for l in [0..n-1]:
    tempdets.append(matrix(2,2,[i,j,k,l]).det().mod(n))
  dets.append(tempdets)
show(table(dets))
  3. You can modify the sage code above to let $n=7, 11, 13,$ etc. How many matrices are in $\text{SL}(2,\mathbb{Z}_7)$ and $\text{GL}(2,\mathbb{Z}_7)$? What patterns did you use to help you count?
  4. If $p$ is a prime, make a conjecture about the order of $\text{SL}(2,\mathbb{Z}_p)$ and the order of $\text{GL}(2,\mathbb{Z}_p)$. You do not have to prove your conjecture.

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