Results 1 to 4 of 4

Math Help - kth power of permutation equals identity element

  1. #1
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1

    kth power of permutation equals identity element

    I have been looking over this cycle and permutation business in an abstract algebra book I have. One of the problems says to prove that the kth power of a permutation equals the identity.

    {\sigma}=(a_{1} \;\ a_{2} \;\ .... \;\ a_{k})\in S_{n}.

    Prove that (a_{1} \;\ a_{2} \;\ ..... \;\ a_{k})^{k}=e

    How do we find the power of a permutation?. i.e (1 \;\ 2 \;\ 3 \;\ 4 \;\ 5)^{3}.

    I can't seem to find it explained anywhere.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,969
    Thanks
    1788
    Awards
    1
    Quote Originally Posted by galactus View Post
    I have been looking over this cycle and permutation business in an abstract algebra book I have. One of the problems says to prove that the kth power of a permutation equals the identity.
    {\sigma}=(a_{1} \;\ a_{2} \;\ .... \;\ a_{k})\in S_{n}.
    Prove that (a_{1} \;\ a_{2} \;\ ..... \;\ a_{k})^{k}=e
    How do we find the power of a permutation?. i.e (1 \;\ 2 \;\ 3 \;\ 4 \;\ 5)^{3}.
    As you well know, notation depends on the author.
    I like the function notation for permutations.
    Say \sigma  = \left( {\begin{array}{ccccc}   1 & 2 & 3 & 4 & 5  \\   3 & 4 & 5 & 1 & 2  \\\end{array} } \right) and we want to find \sigma^3
    We do three compositions.
    \begin{gathered}<br />
  1\xrightarrow{1}3\xrightarrow{2}5\xrightarrow{3}2 \hfill \\<br />
  2\xrightarrow{1}4\xrightarrow{2}1\xrightarrow{3}3 \hfill \\<br />
   \vdots  \hfill \\<br />
  5\xrightarrow{1}2\xrightarrow{2}4\xrightarrow{3}1 \hfill \\ <br />
\end{gathered}

    So \sigma ^3  = \left( {\begin{array}{ccccc}   1 & 2 & 3 & 4 & 5  \\   2 & 3 & 4 & 5 & 1  \\ \end{array} } \right)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    Oh, OK. that's how you do it. Thanks, Plato.
    Last edited by galactus; November 14th 2009 at 05:16 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Eater of Worlds
    galactus's Avatar
    Joined
    Jul 2006
    From
    Chaneysville, PA
    Posts
    3,001
    Thanks
    1
    DUH, I see what is going on now. I was misreading the problem.
    Last edited by galactus; November 14th 2009 at 05:15 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: March 23rd 2010, 08:05 PM
  2. Replies: 2
    Last Post: February 25th 2010, 08:10 PM
  3. identity element of Group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 2nd 2009, 10:23 PM
  4. Identity Element
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 10th 2009, 09:41 PM
  5. Identity Element for Convolution
    Posted in the Calculus Forum
    Replies: 4
    Last Post: August 22nd 2008, 03:22 PM

Search Tags


/mathhelpforum @mathhelpforum