# Thread: Elements in general and special linear groups

1. ## Elements in general and special linear groups

Hi, i am really struggling with this question. There seems to be lots of different methods and it's very confusing! Any help would be greatly appreciated!

Consider the field Fp with p elements. Determine the number of elements in the following groups:
(i) GLn(Fp)
(ii) SLn(Fp)

So far I have (p^n -1)(p^n -p)...(p^n -p^(n-1)) and now I am stuck as my lectures are rather unhelpful!

2. Originally Posted by katiedavies1990
Hi, i am really struggling with this question. There seems to be lots of different methods and it's very confusing! Any help would be greatly appreciated!

Consider the field Fp with p elements. Determine the number of elements in the following groups:
(i) GLn(Fp)
(ii) SLn(Fp)

So far I have (p^n -1)(p^n -p)...(p^n -p^(n-1)) and now I am stuck as my lectures are rather unhelpful!

Well, you already have (i), and for two just note that $\displaystyle SL_n(\mathbb{F}_p)=ker\phi$ , with $\displaystyle \phi:\,GL_n(\mathbb{F}_p)\rightarrow\,\mathbb{F}^{ *}$ defined by $\displaystyle \phi(\alpha):=det(\alpha)$ ...

Tonio

3. Thank ou for your help Tonio, however I have looked up further and still don't understand the special linear group bit. i realise kernel is the same as nullspace but I don't understand what I am trying to prove.
Thanks again,
Katie.

4. Originally Posted by tonio
Well, you already have (i), and for two just note that $\displaystyle SL_n(\mathbb{F}_p)=ker\phi$ , with $\displaystyle \phi:\,GL_n(\mathbb{F}_p)\rightarrow\,\mathbb{F}^{ *}$ defined by $\displaystyle \phi(\alpha):=det(\alpha)$ ...

Tonio
Thank you for your help Tonio, however I have looked up further and still don't understand the special linear group bit. i realise kernel is the same as nullspace but I don't understand what I am trying to prove.
Thanks again,
Katie.

5. Originally Posted by katiedavies1990
Thank you for your help Tonio, however I have looked up further and still don't understand the special linear group bit. i realise kernel is the same as nullspace but I don't understand what I am trying to prove.
Thanks again,
Katie.

No nullspace and stuff: we're in group theory here and you ought to know what the kernel of a group homomorphism is, otherwise this question is out of your league, at least for now.
The special group is just the subgroup of all the invertible matrices whose determinant is 1...
Now check again my first message to you and use the first isomorphism theorem for group homomorphisms...and perhaps Lagrange's Theorem, too.

Toio