Results 1 to 7 of 7

Thread: Groups-Permutation groups

  1. #1
    Member Jason Bourne's Avatar
    Joined
    Nov 2007
    Posts
    132

    Groups-Permutation groups

    I have the following questions but I don't totally understand them, so im not sure about it all:

    Let H be the subgroup of $\displaystyle S_8$ generated by the permutations

    $\displaystyle \alpha = (1 2 3 4)(5 6 7 8), \beta = (1 5 3 7)(2 8 4 6)
    $

    (1) Find the order of $\displaystyle \alpha, \alpha\beta$. What's the order of $\displaystyle \beta^{-1}\alpha\beta$ ?

    (2) Find the order of H.

    (3) Find the conjugacy classes of H.

    (4) Show that $\displaystyle Z=\{e, \alpha^2 \}$ is a normal subgroup of H

    (5) To which well-known group is the quotient H/Z isomorphic?
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    For (1) I know the orders are all 4 I think.

    For (2) Is the fastest way of doing this to actually go through and compute every new element? Or is there an easier way of finding the order?

    For (3) I think the conjugacy classes are the sets in which the permutions all have the same cycle structure?

    For (4) Z is normal iff Z is a union of conjugacy classes?

    For (5) I think this is isomorphic to the Klein Four Group. Find the orders of elements in H/Z ?

    Thanks for any help.
    Last edited by Jason Bourne; May 12th 2009 at 04:18 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie adit38's Avatar
    Joined
    May 2009
    From
    Praya - Lombok - Indonesia
    Posts
    3
    i am nebie....., but i enjoy grup theory, can i follow this thread?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Jason Bourne's Avatar
    Joined
    Nov 2007
    Posts
    132
    Quote Originally Posted by adit38 View Post
    i am nebie....., but i enjoy grup theory, can i follow this thread?
    Yes you can follow this thread and feel free to join in.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Jason Bourne View Post
    I have the following questions but I don't totally understand them, so im not sure about it all:

    Let H be the subgroup of $\displaystyle S_8$ generated by the permutations

    $\displaystyle \alpha = (1 2 3 4)(5 6 7 8), \beta = (1 5 3 7)(2 8 4 6)
    $

    (1) Find the order of $\displaystyle \alpha, \alpha\beta$. What's the order of $\displaystyle \beta^{-1}\alpha\beta$ ?

    (2) Find the order of H.

    (3) Find the conjugacy classes of H.

    (4) Show that $\displaystyle Z=\{e, \alpha^2 \}$ is a normal subgroup of H

    (5) To which well-known group is the quotient H/Z isomorphic?
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    For (1) I know the orders are all 4 I think.

    For (2) Is the fastest way of doing this to actually go through and compute every new element? Or is there an easier way of finding the order?

    For (3) I think the conjugacy classes are the sets in which the permutions all have the same cycle structure?

    For (4) Z is normal iff Z is a union of conjugacy classes?

    For (5) I think this is isomorphic to the Klein Four Group. Find the orders of elements in H/Z ?

    Thanks for any help.
    this is a long exercise ... here are some hints:

    1. In any group, the order of an element and any of its conjugates are equal because $\displaystyle (y^{-1}xy)^k=y^{-1}x^ky.$ so $\displaystyle o(\beta^{-1} \alpha \beta)=o(\alpha)=4.$

    2. show that $\displaystyle \alpha^2=\beta^2$ and $\displaystyle \alpha \beta \alpha = \beta$ and therefore $\displaystyle H \cong Q_8,$ the quaternion group of order 8.

    3. $\displaystyle H$ has 5 conjugacy classes: $\displaystyle \{1 \}, \ \{\alpha, \alpha^3 \}, \ \{\alpha^2 \}, \ \{\beta, \beta^3 \}, \ \{\alpha \beta, \alpha \beta^3 \}.$
    Last edited by NonCommAlg; May 12th 2009 at 12:22 PM. Reason: 5 not 4.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member Jason Bourne's Avatar
    Joined
    Nov 2007
    Posts
    132
    2. show that $\displaystyle \alpha^2=\beta^2$ and $\displaystyle \alpha \beta \alpha = \beta$ and therefore $\displaystyle H \cong Q_8,$ the quaternion group of order 8.
    I don't understand how this implies H is isomorphic to Q8

    3. $\displaystyle H$ has 4 conjugacy classes: $\displaystyle \{1 \}, \ \{\alpha, \alpha^3 \}, \ \{\alpha^2 \}, \ \{\beta, \beta^3 \}, \ \{\alpha \beta, \alpha \beta^3 \}.$
    I would have though that the conjugacy classes are:

    $\displaystyle \{1 \},\ \{\alpha^2 \}, \ \{\alpha, \alpha^3 , \beta, \beta^3, \alpha \beta, \alpha \beta^3 \}$

    because the elements in each set have the same disjoint cycle structure?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by Jason Bourne View Post

    I don't understand how this implies H is isomorphic to Q8
    because they have the same presentations.


    I would have though that the conjugacy classes are:

    $\displaystyle \{1 \},\ \{\alpha^2 \}, \ \{\alpha, \alpha^3 , \beta, \beta^3, \alpha \beta, \alpha \beta^3 \}$

    because the elements in each set have the same disjoint cycle structure?
    those are conjugates in $\displaystyle S_8$ not in $\displaystyle H.$ don't forget that we want to find the conjugacy classes of $\displaystyle H$.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member Jason Bourne's Avatar
    Joined
    Nov 2007
    Posts
    132
    because they have the same presentations.
    Presentations????

    those are conjugates in $\displaystyle S_8$ not in $\displaystyle H.$ don't forget that we want to find the conjugacy classes of $\displaystyle H$.
    Now im confused. So how do I find the conjugacy classes of H? I'm clearly not understanding this stuff.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Permutation of groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 25th 2011, 01:13 AM
  2. Permutation Groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Sep 3rd 2010, 04:04 PM
  3. Permutation groups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Nov 25th 2008, 12:56 AM
  4. Permutation Groups
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Oct 4th 2008, 05:14 PM
  5. Permutation Groups...
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 24th 2007, 08:55 PM

Search Tags


/mathhelpforum @mathhelpforum