# Elements in general and special linear groups

• Dec 3rd 2009, 03:17 AM
katiedavies1990
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! (Happy)

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!

• Dec 3rd 2009, 03:31 AM
tonio
Quote:

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! (Happy)

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 $SL_n(\mathbb{F}_p)=ker\phi$ , with $\phi:\,GL_n(\mathbb{F}_p)\rightarrow\,\mathbb{F}^{ *}$ defined by $\phi(\alpha):=det(\alpha)$ ...

Tonio
• Dec 6th 2009, 07:17 AM
katiedavies1990
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. (Worried)
Thanks again,
Katie.
• Dec 6th 2009, 08:19 AM
katiedavies1990
Quote:

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

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. (Worried)
Thanks again,
Katie. http://www.mathhelpforum.com/math-he...c/progress.gif
• Dec 6th 2009, 08:33 AM
tonio
Quote:

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. (Worried)
Thanks again,
Katie. http://www.mathhelpforum.com/math-he...c/progress.gif

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