Results 1 to 4 of 4
Like Tree2Thanks
  • 1 Post By ILikeSerena
  • 1 Post By Nehushtan

Math Help - Permutations

  1. #1
    Junior Member
    Joined
    Mar 2010
    Posts
    25

    Permutations

    Hey, i'm having a hard time seeing how to do this problem. Any hints?

    \alpha \in S_{n} a cycle   (a_{1},a_{2},...,a_{k}) (a_{1},a_{2},...,a_{k} \:distinct \:elements \:from \:the \:set \:\{1,2,..n\}). Let \beta  \in S_{n} a permutation. Show that

    \beta\alpha \beta ^{-1} = (\beta (a_{1}),\beta (a_{2}),...,\beta (a_{k})).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member ILikeSerena's Avatar
    Joined
    Dec 2011
    Posts
    733
    Thanks
    121

    Re: Permutations

    Quote Originally Posted by gordo151091 View Post
    Hey, i'm having a hard time seeing how to do this problem. Any hints?

    \alpha \in S_{n} a cycle   (a_{1},a_{2},...,a_{k}) (a_{1},a_{2},...,a_{k} \:distinct \:elements \:from \:the \:set \:\{1,2,..n\}). Let \beta  \in S_{n} a permutation. Show that

    \beta\alpha \beta ^{-1} = (\beta (a_{1}),\beta (a_{2}),...,\beta (a_{k})).
    Hi gordo151091!

    Can you apply \beta\alpha \beta^{-1} to \beta (a_{1})?
    What do you get?
    Thanks from gordo151091
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member Nehushtan's Avatar
    Joined
    Mar 2013
    From
    Europe
    Posts
    42
    Thanks
    12

    Re: Permutations

    Consider the effect of \beta\alpha\beta^{-1} on \beta(a_i) for i=1,\ldots,k-1. We have

    \beta\alpha\beta^{-1}(\beta(a_i))=\beta\alpha(a_i)=\beta(a_{i+1})

    So \beta\alpha\beta^{-1} maps \beta(a_i) to \beta(a_{i+1}) for i=1,\ldots,k-1, and it maps \beta(a_k) to \beta(a_1) since \alpha maps a_k to a_1.

    Now suppose b\in\{1,\ldots,n\} and b\notin\{\beta(a_1),\ldots,\beta(a_k)\}. Then b=\beta(a) for some a\notin\{a_1,\ldots,a_k\}. Thus \alpha fixes a and

    \beta\alpha\beta^{-1}(b)=\beta\alpha\beta^{-1}(\beta(a))=\beta\alpha(a)=\beta(a)=b

    so \beta\alpha\beta^{-1} fixes b.

    Hence \beta\alpha\beta^{-1}=(\beta(a_1)\,\beta(a_2)\,\cdots\,\beta(a_k)).
    Thanks from gordo151091
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Mar 2010
    Posts
    25

    Re: Permutations

    Hi! Thanks to you two for your help
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Permutations in S_6
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 5th 2011, 01:46 AM
  2. Permutations
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: December 6th 2010, 06:52 PM
  3. How many permutations are there?
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: March 22nd 2010, 03:37 PM
  4. permutations
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: February 11th 2009, 06:42 PM
  5. Permutations
    Posted in the Statistics Forum
    Replies: 1
    Last Post: September 20th 2008, 12:11 PM

Search Tags


/mathhelpforum @mathhelpforum