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Math Help - Abelian Midterm question.

  1. #1
    Member elninio's Avatar
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    Abelian Midterm question.

    I had never encountered this type of question until a couple of minutes ago during a midterm...

    It went something like this

    Prove that If x^3=x for all x in G, then G is abelian.

    I stated that since x^3=x, the only possibilities for x are 1,0 and -1. In this case I proved that all three combinations of these numbers are abelian.

    Looking through some notes, should I have introduced a y with the same properties and then proven something like x^3 * y^3 =xy?

    How is this proof done?
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  2. #2
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    Quote Originally Posted by elninio View Post
    I had never encountered this type of question until a couple of minutes ago during a midterm...

    It went something like this

    Prove that If x^3=x for all x in G, then G is abelian.

    I stated that since x^3=x, the only possibilities for x are 1,0 and -1. In this case I proved that all three combinations of these numbers are abelian.

    Looking through some notes, should I have introduced a y with the same properties and then proven something like x^3 * y^3 =xy?

    How is this proof done?
    You don't even know that x is a number -- so you can't say that x^3=x means x=0,\pm 1...

    The correct way to go about it would be to note that x^3=x \ \forall x \in G \Rightarrow x^2 = e \ \forall x \in G, where e is the identity element in G. Now, I assume you proved the theorem that states that if every element of G is of order 2 (other than the identity obviously), then G is abelian... if you haven't, then prove that and you are done.
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