Question

Compute the order of SL2(Zp) and explain why

Answer 1

SL₂(Z_p) is the set of all invertible square matrices of order 2 whose determinant is 1 and the elements of the matrices are from the field Z_p , where p is necessarily a prime.

First select the elements of the main diagonal.

There p−1 ways to select them so that their product is 1.

If the product is 1 the product of the elements of the other diagonal must be 0. There are 2p−1ways to select them so at least one of them is a multiple of p.

This gives us (p−1)(2p−1) matrices.

There are p²−p+1 selections in which the product of the diagonal is not 1.

No matter what the product of the diagonal is, the product for the other diagonal shall be fixed, and will be non-zero. There are p−1 ways to choose them so that they give the selected product.

So we have (p−1)(2p−1)+(p²−p+1)(p−1) = (p−1)(p²+p) = p³−p(p−1)(2p−1)+(p²−p+1)(p−1) = (p−1)(p²+p) = p³−p such matrices.

hence, |SL₂(Z_p)| = p³−p