Results 1 to 5 of 5

Math Help - Elements in general and special linear groups

  1. #1
    Newbie
    Joined
    Dec 2009
    Posts
    5

    Exclamation 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!

    Thanks in advance, Katie

    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by katiedavies1990 View Post
    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
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2009
    Posts
    5
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Dec 2009
    Posts
    5
    Quote Originally Posted by tonio View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by katiedavies1990 View Post
    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
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Groups of 44 elements
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 3rd 2011, 02:38 PM
  2. Replies: 7
    Last Post: October 10th 2011, 03:06 PM
  3. [SOLVED] Groups of 3,4,5 elements abelian
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: May 17th 2010, 02:30 PM
  4. elements of symmetric groups and centralizers
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 11th 2010, 02:15 AM
  5. Orders of elements in Symmetric groups
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: March 30th 2010, 02:39 PM

Search Tags


/mathhelpforum @mathhelpforum