Results 1 to 5 of 5

Math Help - composition of permutation groups

  1. #1
    fzr
    fzr is offline
    Newbie
    Joined
    Dec 2007
    Posts
    3

    Question composition of permutation groups

    hi, could anyone explain to me how you combine those? i just don't get it..
    for example:
    (1 2 3) o (1 2 3) = (1 2 3)
    (2 1 3)__(3 2 1)___(3 1 2)
    or
    (p q r) o (p q r) = (p q r)
    (q r p)__(r p q)___(p q r)
    (ignore the underscores)
    thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,803
    Thanks
    1692
    Awards
    1
    You know how function composition works.
    If f(x) = x^2 \,\& \,c(x) = \cos (x) then f \circ c(x) = \cos ^2 (x)\,\& \,c \circ f(x) = \cos \left( {x^2 } \right).
    The “inter” function operates first.
    So if p = \left( {\begin{array}{*{20}c}    1 & 2 & 3 & 4  \\    3 & 2 & 4 & 1  \\<br />
\end{array}} \right)\,\& \,q = \left( {\begin{array}{*{20}c}    1 & 2 & 3 & 4  \\<br />
   2 & 4 & 1 & 3  \\ \end{array}} \right) then in p \circ q the q permutation acts first and then the p acts.
    Thus we have p \circ q = \left( {\begin{array}{*{20}c}    1 & 2 & 3 & 4  \\<br />
   2 & 1 & 3 & 4  \\ \end{array}} \right)\,\& \,q \circ p = \left( {\begin{array}{*{20}c}   1 & 2 & 3 & 4  \\   1 & 4 & 3 & 2  \\\end{array}} \right)\,
    Last edited by Plato; December 16th 2007 at 10:25 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    fzr
    fzr is offline
    Newbie
    Joined
    Dec 2007
    Posts
    3
    so when you combine P and Q you have
    1->2 followed by 4->1 gives us 1->2
    2->4 followed by 3->4 gives us 2->1
    3->1 followed by 2->2 gives us 3->3
    4->3 followed by 1->3 gives us 4->4
    is that right? but I just dont get it how you end up with those numbers. do you multiply, add, subtract or do something else?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,803
    Thanks
    1692
    Awards
    1
    Quote Originally Posted by fzr View Post
    so when you combine P and Q you have
    1->2 followed by 4->1 gives us 1->2
    2->4 followed by 3->4 gives us 2->1
    3->1 followed by 2->2 gives us 3->3
    4->3 followed by 1->3 gives us 4->4
    is that right? but I just dont get it how you end up with those numbers. do you multiply, add, subtract or do something else?
    NO, not that at all.
    When we do p \circ q we get
    1->2 followed by 2->2 gives us 1->2
    2->4 followed by 4->1 gives us 2->1
    3->1 followed by 1->3 gives us 3->3
    4->3 followed by 3->4 gives us 4->4.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    fzr
    fzr is offline
    Newbie
    Joined
    Dec 2007
    Posts
    3
    oh now i get it... thanks m8
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Permutation Groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 3rd 2010, 04:04 PM
  2. Permutation Groups
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: August 15th 2010, 01:47 AM
  3. Permutation groups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 25th 2008, 12:56 AM
  4. Permutation Groups
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 4th 2008, 05:14 PM
  5. Permutation Groups...
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 24th 2007, 08:55 PM

Search Tags


/mathhelpforum @mathhelpforum