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Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of order2

  1. Apr 6, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that a non-abelian group of order 10 must have an element of order 2.



    What if the order of every element is 5?
    Prove there are 5 elements of order 2.
     
  2. jcsd
  3. Apr 6, 2009 #2
    Re: Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of or

    Where is your working? Have you attempted this?
     
  4. Apr 6, 2009 #3
    Re: Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of or

    honestly, I have no work because I don't know where to begin
     
  5. Apr 6, 2009 #4
    Re: Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of or

    I don't know what you mean by this but for the first part use Lagrange's theorem deduce that there are 2 possible orders of elements. If you assume there is no element of order 2 prove that this means the group is abelian.
     
  6. Apr 6, 2009 #5
    Re: Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of or

    sorry,
    it is all one problem


    Prove that a nonabelian group of order 10 must have an element of order 2. What if the order of every element is 5? Prove there are 5 elements of order 2.


    I am having trouble understanding Lag. THM.

    Thanks again for your help
     
  7. Apr 6, 2009 #6
    Re: Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of or

    Lagranges theorem says that the order of the subgroup must divide the order of the group. The order of a cyclic group is prime. If you take any element of the group, you can make a cyclic subgroup generated by that element, so Lagrange says that the order of any element must divide the order of the group. The two possibilities you have for a non identity element are 2 and 5.
     
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