Order of SL(2,Zp), p is a prime.

[Silversonic]

Homework Statement

Determine the index [G], where H is a subgroup of the group G and;

G = GL(2,Z_p)

H = SL(2,Z_p)

p is a prime

Where GL(2,Z_p) is the general linear group of 2x2 invertible matrices with entries in Z_p, SL(2,Z_p) is the general linear group of 2x2 invertible matrices with entries in Z_p but with determinant 1.

Homework Equations

|GL(2,Z_p)| = (p^2 -1)(p^2 -p)

Lagrange's theorem

|GL(2,Z_p)| = [G<img src="/styles/physicsforums/xenforo/smilies/arghh.png" class="smilie" loading="lazy" alt=":H" title="Gah! :H" data-shortname=":H" />]|SL(2,Z_p)|

The Attempt at a Solution

To be able to do this, I need to intuitively work out the number of distinct cosets, or work out the order of SL(2,Z_p) in terms of p.

I know how to derive the order of GL(n,Z_p) (note: not necessarily a 2x2 matrix), but doing the same for the special linear group doesn't seem to be as straightforward.

I've looked at it this way, the determinant of SL(2,Z_p) must be one, so

ac-bd = 1 - the determinant

Thus if we let

ac = m

Then

bd = m - 1

1 \leq m \leq p^2

So, as a or c cannot be zero, a and c can have values ranging from 1 to p-1. i.e. a can have (p-1) different values and so can c. So the number of matrices in SL(2,Z_p) is

(Number of combinations of a) times (Number of combinations of c) times (Number of combinations of b) times (Number of combinations of d)

=

(p-1) times (p-1) times (Number of combinations of b) times (Number of combinations of d)

=

(p-1)^2 times (Number of combinations of b) times (Number of combinations of d)

How would I work out the number of possible combinations of b and d from this? Once I work that out, I should times (p-1)^2 by those to get the order of SL(2,Z_p).

 

[micromass]

Hmmm, what if you apply the isomorphism theorem to the map

GL(2,\mathbb{Z}_p)\rightarrow \mathbb{Z}_p:A\rightarrow det(A)

??