Results 1 to 4 of 4

Thread: Order of a group

  1. #1
    Senior Member vincisonfire's Avatar
    Joined
    Oct 2008
    From
    Sainte-Flavie
    Posts
    468
    Thanks
    2
    Awards
    1

    Order of a group

    Let $\displaystyle \mathbb F $ be a field. Prove that $\displaystyle SL_2 (\mathbb F) $ be the group of 2 x 2 matrices with coefficient in $\displaystyle \mathbb F $ and determinant equal to 1.
    Question : If F has q elements, how many elements are in the group $\displaystyle SL_2 (\mathbb F) $?
    I know that $\displaystyle |M_2(\mathbb F)| = q^4$.
    $\displaystyle SL_2 (\mathbb F) $ is a subgroup of $\displaystyle GL_2 (\mathbb F) $ the group of units of $\displaystyle M_2(\mathbb F) $.
    From this I've tried to solve the problem but I'm not able to find a way to count the elements in any of the groups.
    Last edited by vincisonfire; Nov 30th 2008 at 03:52 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by vincisonfire View Post
    Let $\displaystyle \mathbb F $ be a field. Prove that $\displaystyle SL_2 (\mathbb F) $ be the group of 2 x 2 matrices with coefficient in $\displaystyle \mathbb F $ and determinant equal to 1.
    Question : If F has q elements, how many elements are in the group $\displaystyle SL_2 (\mathbb F) $?
    Hint: $\displaystyle \text{GL}_2(\mathbb{F})/\text{SL}_2(\mathbb{F}) \simeq \mathbb{F}^{\times}$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member vincisonfire's Avatar
    Joined
    Oct 2008
    From
    Sainte-Flavie
    Posts
    468
    Thanks
    2
    Awards
    1
    $\displaystyle |\text{GL}_2(\mathbb{F})/\text{SL}_2(\mathbb{F})| = |\mathbb{F}^{\times}| = q-1 $ because any element except 0 has a multiplicative inverse in $\displaystyle \mathbb F $.
    In other word, there are $\displaystyle q-1 $ equivalent classes (or orbits) of $\displaystyle \text{SL}_2(\mathbb{F}) $.
    We thus need to find the number of elements in each orbit.
    In an equivalent class we have
    $\displaystyle A=\left( \begin{array}{cc}q&a\\0&q^{-1} \end{array} \right) $ $\displaystyle A \in \text{SL}_2(\mathbb{F}) $
    There are $\displaystyle \frac{q}{2} $combinations of $\displaystyle q \text{ and } q^{-1} $.
    $\displaystyle 2q $combinations of $\displaystyle a $because we can swap $\displaystyle a \text{ and } 0 $.
    There are therefore $\displaystyle q^2 $elements in its equivalent classes.
    $\displaystyle |\text{SL}_2(\mathbb{F})| = q^2(q-1) = q^3-q $
    Am I right?
    Last edited by vincisonfire; Nov 30th 2008 at 04:54 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by vincisonfire View Post
    $\displaystyle |\text{GL}_2(\mathbb{F})/\text{SL}_2(\mathbb{F})| = |\mathbb{F}^{\times}| = q-1 $ because any element except 0 has a multiplicative inverse in $\displaystyle \mathbb F $.
    In other word, there are $\displaystyle q-1 $ equivalent classes (or orbits) of $\displaystyle \text{SL}_2(\mathbb{F}) $.
    We thus need to find the number of elements in each orbit.
    In an equivalent class we have
    $\displaystyle A=\left( \begin{array}{cc}q&a\\0&q^{-1} \end{array} \right) $ $\displaystyle A \in \text{SL}_2(\mathbb{F}) $
    There are $\displaystyle \frac{q}{2} $combinations of $\displaystyle q \text{ and } q^{-1} $.
    $\displaystyle 2q $combinations of $\displaystyle a $because we can swap $\displaystyle a \text{ and } 0 $.
    There are therefore $\displaystyle q^2 $elements in its equivalent classes.
    $\displaystyle |\text{SL}_2(\mathbb{F})| = q^2(q-1) = q^3-q $
    Am I right?
    $\displaystyle | \text{GL}_2(\mathbb{F}) | = (q^2 - 1)(q^2 - q)$

    Therefore, $\displaystyle \text{SL}_2(\mathbb{F}) = \frac{\text{GL}_2(\mathbb{F})}{q-1} = \frac{(q^2-1)(q^2 - q)}{q-1} = (q-1)q(q+1)$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Order of Group. Direct Product of Cyclic Group
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: Nov 19th 2011, 01:06 PM
  2. Order of a Group, Order of an Element
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Nov 15th 2010, 06:28 PM
  3. Order of a Group, Order of an Element
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Nov 15th 2010, 06:02 PM
  4. Prove that a group of order 375 has a subgroup of order 15
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 13th 2010, 11:08 PM
  5. Group of order pq must have subgroups of order p and q.
    Posted in the Advanced Algebra Forum
    Replies: 19
    Last Post: Sep 15th 2009, 12:04 PM

Search Tags


/mathhelpforum @mathhelpforum