Results 1 to 2 of 2

Math Help - Monoids and Groups

  1. #1
    Junior Member
    Joined
    Nov 2008
    Posts
    36

    Monoids and Groups

    Can I get help on the following problem:
    Show that if n GE 3 then the center of Sn is of order 1.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by thomas_donald View Post
    Can I get help on the following problem:
    Show that if n GE 3 then the center of Sn is of order 1.
    Let n\geq 4.

    Let \sigma be non-trivial permutation. Then there exists a,b\in \{1,2,...,n\} such that \sigma(a) = b with a\not = b. Notice that a,\sigma^{-1}(a),b has at most three distinct points, let c be different from all three of these (since n\geq 4 this is possible). There exists a permutation \tau which satisfies \tau(a) = \sigma^{-1}(a),\tau(b) = c. Thus, \tau \sigma(a) = \tau (b) = c and \sigma \tau(a) = \sigma \sigma^{-1}(a) = a. We see that \tau \sigma \not = \sigma \tau. This shows if \sigma is not trivial then it cannot lie in the center.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Quotient Groups - Infinite Groups, finite orders
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: August 11th 2010, 07:07 AM
  2. Monoids and units help
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 1st 2009, 08:02 PM
  3. Monoids with two elements
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 25th 2009, 09:33 PM
  4. Monoids and Groups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: September 4th 2009, 10:21 AM
  5. Order of groups involving conjugates and abelian groups
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: February 5th 2009, 08:55 PM

Search Tags


/mathhelpforum @mathhelpforum