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Show there exist an element of order 2 in this group

  1. Dec 4, 2009 #1
    1. The problem statement, all variables and given/known data
    If G is a finite groups whose order is even, then there exists an element a in G whose order is 2.


    2. Relevant equations



    3. The attempt at a solution
    does this mean that a^2 is the identity? how can i prove this? Also, would't this make G cyclic?
     
  2. jcsd
  3. Dec 4, 2009 #2
    Re: groups...

    Yes, that would make a2 = 1
    Perhaps you should check out Cauchy's Theorem
     
  4. Dec 5, 2009 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Re: groups...

    Yes, that means that [itex]a^2[/itex] is the identity. And that, in turn, means that a is its own inverse.

    The group identity has the property that it is its own inverse (but has order 1, not 2). Suppose that there were no other member of the group with that property. Then we could pair the other members of the group, each paired with its inverse.
     
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