Results 1 to 2 of 2

Math Help - more about finite groups

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    5

    more about finite groups

    Here is another problem I struggled with about finite groups. Thanks!

    1. Let G be a finite group. Show that the number of elements, x, in G such that x^3=1 is odd.

    Thanks!
    Last edited by mr fantastic; April 30th 2009 at 02:50 PM. Reason: Restored original question and closed thread
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Gamma's Avatar
    Joined
    Dec 2008
    From
    Iowa City, IA
    Posts
    517

    Order 3

    The only things that this will be true for are things with order dividing 3. That is order 1 and 3. The identity is the only thing with order 1.

    I don't know what you guys have proved, but all of these are elementary to prove, so I will leave it to you.

    Inverses are unique.
    The order of an element is the same as the order of it's inverse.
    Anything that is it's own inverse has order 2.

    The point is, for every element in G with order 3, it has a unique counterpart (it's inverse) that also has order 3. You don't have to worry about double counting something because an element cannot be it's own inverse and still have order 3. So that means there must be an even number of things with order 3. Then you add in the identity which has order 1 and will still satisfy the requirement. So this is an odd number of elements in the group which satisfy x^3=1.

    Let me know if any of those 3 observations are not clear to you, but I think you can check those things by just following your nose.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. finite abelian groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 2nd 2010, 09:00 PM
  2. Quotient Groups - Infinite Groups, finite orders
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: August 11th 2010, 08:07 AM
  3. Finite Abelian Groups
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 8th 2009, 04:12 PM
  4. result about finite groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 10th 2009, 07:28 PM
  5. residually finite groups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 18th 2009, 07:05 PM

Search Tags


/mathhelpforum @mathhelpforum