Elements in general and special linear groups

**katiedavies1990** · December 3rd 2009, 02:17 AM

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!

Thanks in advance, Katie

**tonio** · December 3rd 2009, 02:31 AM

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!

Thanks in advance, Katie

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

**katiedavies1990** · December 6th 2009, 06:17 AM

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.

**katiedavies1990** · December 6th 2009, 07:19 AM

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.

Thanks again,
Katie.

**tonio** · December 6th 2009, 07:33 AM

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

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