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Homework Help: Group Theory simple question

  1. Feb 13, 2015 #1
    1. The problem statement, all variables and given/known data
    If G is a group of even order, show that it has an element g not equal to the identity such that g^2 = 1.

    2. Relevant equations

    3. The attempt at a solution
    What I wrote:

    If |G| = n, then g^n = 1 for some g in G. Thus, (g^(n/2))(g^(n/2)) = 1, so g^(n/2) is the element of order 2.

    Is this a flawed argument? there guaranteed to be an element such that g^n = 1?
  2. jcsd
  3. Feb 13, 2015 #2


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    The identity is, of course, its own inverse. Since the group is "of even order", i.e. it has an even number of elements, removing the identity leaves an odd number of elements. Pairing each number with its inverse, what happens?
  4. Feb 15, 2015 #3
    I see your arguement. Is this true though? Can't ab = 1, bc = 1, cd = 1 etc etc but ba not equal 1?
  5. Feb 15, 2015 #4


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    Homework Helper

    No, it's not possible. If ab=1 then ba=1. Prove it!
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