Results 1 to 3 of 3
Like Tree1Thanks
  • 1 Post By johnsomeone

Math Help - Order of a Group and its Element(s)

  1. #1
    Newbie
    Joined
    Sep 2011
    Posts
    11

    Order of a Group and its Element(s)

    Let G be a group of order 6. Show that if G is not abelian then G has an element of order 3.

    I believe that if g, h \in G then  gh \neq hg in general, and this means that G cannot be a cyclic. Also  g^6 = 1 as a corollary of Lagrange's Theorem. I'm not sure if any of this is helpful to the problem at hand though... I'm quite stuck as you can see.

    Thanks,
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    147

    Re: Order of a Group and its Element(s)

    You're on the right track. You've shown that it can't be cyclic, because cyclic groups are abelian.
    The way I see is to proceed to show it by contradiction. Assume G has no element of order 3.
    Since G isn't cyclic, G has no element of order 6.
    By assumption, G has no element of order 3.
    With that information, and |G| = 6, you can determine the orders of G's elements (hint: Lagrange.)
    Once you've done that, consider the two elements in G, a and b, that don't commute.
    That means that ba is not equal to ab.
    The contradiction will appear by looking at g = ab and asking what happens when you square g.
    Remember that you'll know something about the order of g at this point - and also the orders of a and b.
    So consider what you can say about g squared = abab for those non-commuting a and b.
    With a little bit of manipulation when considering g squared, the contradiction will appear.
    (The contraction will arise by showing that a and b do commute.)
    Thus the assumption was false, and so G has an element of order 3.
    Last edited by johnsomeone; October 19th 2012 at 07:04 PM.
    Thanks from anguished
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2011
    Posts
    11

    Re: Order of a Group and its Element(s)

    Ah, excellent. When you describe it like that, I see how the proof can be done. Thanks for the help!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Definition of the order of an element in a group
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: April 11th 2012, 09:43 AM
  2. order of an element of a group
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 30th 2010, 01:55 PM
  3. Group Theory - order of element
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: December 2nd 2009, 05:50 AM
  4. group element order
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 1st 2009, 08:23 AM
  5. order of an element in mulitpicative group
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: February 15th 2009, 01:25 PM

Search Tags


/mathhelpforum @mathhelpforum