Results 1 to 3 of 3

Math Help - order 171

  1. #1
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347

    order 171

    This question is part of a larger question, but I don't understand the solution I have been given here.

    Show that there is no element of order 171 in the symmetric group  S_{20}.

    Solution: The smallest n such that S_n contains an element of order 171 = 3^2 \cdot 19 is n = 9 + 19 = 28.


    I don't understand at all why that statement is true. The question comes at the end of a section about the Sylow theorems. Any help with this would be appreciated
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by slevvio View Post
    This question is part of a larger question, but I don't understand the solution I have been given here.

    Show that there is no element of order 171 in the symmetric group  S_{20}.

    Solution: The smallest n such that S_n contains an element of order 171 = 3^2 \cdot 19 is n = 9 + 19 = 28.


    I don't understand at all why that statement is true. The question comes at the end of a section about the Sylow theorems. Any help with this would be appreciated

    Facts you must check you know/you can prove:

    1) Every permutation can be written as a product of disjoint cycles

    2) The order of a permutation is the lowest common multiple of the orders (=lengths) of the cycles appearing in its decomposition in disjoint cycles.

    From here the solution follows at once.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    Suppose there exists an element of order 171 in  S_n . We want to find the different ways to write this element as the product of disjoint cycles, and then look at the smallest possible one.

    The factors of 171 are 1,3,9,19,57, and 171.

    The smallest one we can get is a cycle of 9 elements times a cycle with 19 elements (lcm(19,9) = 171) , so there must be at least 19 + 9 = 28 elements to choose from to put into those cycles
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Re-writing higher order spatial derivatives as lower order system
    Posted in the Differential Equations Forum
    Replies: 11
    Last Post: July 27th 2010, 09:56 AM
  2. Replies: 1
    Last Post: October 27th 2009, 05:03 AM
  3. Replies: 2
    Last Post: February 23rd 2009, 06:54 AM
  4. Replies: 2
    Last Post: November 25th 2008, 10:29 PM
  5. Replies: 1
    Last Post: May 11th 2007, 04:01 AM

Search Tags


/mathhelpforum @mathhelpforum