# field

• Nov 26th 2006, 04:37 PM
PvtBillPilgrim
field
I have a question that I don't know the answer to.

Let F be a field. I showed that SL2(F) defined to be the matrix
(a b
c d) where a,b,c,d are in F and ad-bc=1 is a group.

However, the question goes on to ask,
If F has q elements, how many elements are in the group SL2(F)?
• Nov 26th 2006, 06:07 PM
ThePerfectHacker
Quote:

Originally Posted by PvtBillPilgrim
I have a question that I don't know the answer to.

Let F be a field. I showed that SL2(F) defined to be the matrix
(a b
c d) where a,b,c,d are in F and ad-bc=1 is a group.

However, the question goes on to ask,
If F has q elements, how many elements are in the group SL2(F)?

Think about it, what you actually want to know are the elements that are units. Our good friend rgep has given an exlanation here.
• Nov 26th 2006, 06:17 PM
PvtBillPilgrim
So it would just be:
((q^n)-1)*((q^n)-(q^n-1)) where n=2 in this case?

Thus, ((q^2)-1)*((q^2)-(q))
• Nov 26th 2006, 06:23 PM
ThePerfectHacker
Quote:

Originally Posted by PvtBillPilgrim
So it would just be:
((q^n)-1)*((q^n)-(q^n-1)) where n=2 in this case?

Thus, ((q^2)-1)*((q^2)-(q))

No!

$(q-1)(q^2-q)(q^3-q^2)....(q^n-q^{n-1})$
Or if you want to be cool :cool:

$\prod_{k=1}^n (q^{k}-q^{k-1})$
• Nov 26th 2006, 06:31 PM
PvtBillPilgrim
I don't know the dimension?