Finding the Class Equation of SL2(F3)

[Silviu]

Hello! Is there a way to find the class equation of SL2(F3), other than writing down all the 24 matrices and do all the necessary computations?

 

[jvicens]

Hello there! Yes, there is a more efficient way to find the class equation of SL2(F3). First, let's define what SL2(F3) is. It is the special linear group of 2x2 matrices over the finite field F3, which has 3 elements.

To find the class equation, we can use the fact that the class equation of a group is equal to the sum of the number of elements in each conjugacy class. In other words, we need to find the number of distinct conjugacy classes in SL2(F3).

To do this, we can use the concept of the centralizer. The centralizer of an element x in a group G is the set of elements in G that commute with x. In other words, it is the set {g∈G | gx=xg}.

Now, in order to find the number of distinct conjugacy classes, we need to find the number of distinct centralizers. This can be done by considering the possible values of the determinant of the 2x2 matrices in SL2(F3). The determinant must be 1 for the matrix to be in SL2(F3).

There are three possible values for the determinant: 1, 0, and -1.

1. When the determinant is 1, we have the identity matrix, which has a centralizer of size 1.
2. When the determinant is 0, we have a matrix with a zero column or row, which cannot have a centralizer of size greater than 1.
3. When the determinant is -1, we have two possible cases: a) the matrix has two identical rows or columns, which again cannot have a centralizer of size greater than 1, or b) the matrix has two distinct rows or columns, which has a centralizer of size 3.

Therefore, we have 1+6+3=10 distinct centralizers, and thus 10 distinct conjugacy classes. This means that the class equation of SL2(F3) will have 10 terms.

I hope this helps! Let me know if you have any further questions.