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Math Help - commutativity of a group

  1. #1
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    commutativity of a group

    I'm trying to prove this problem but I'm not sure whether or not my answer is working. The problem states

    "Let G be a group, with the following property: Whenever a, b, and c belong to G and ab=ca, then b=c. Prove that G is abelian."

    I have solved for b and c using inverses, and got that b=a-1(ca) and c=(ab)a-1. The second half of the problem states that b=c. So I wanted to use replacement to show that the group is abelian. But I'm not sure if that would prove G is abelian for any arbitrary b and c or if its only commutative for this exact pair.
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  2. #2
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    Hi

    Perhaps it is better to start with two elements, let's say a,b, in G. You want to prove ab=ba.

    Inverses are indeed often useful for group equations: take a look at ab=ab(a^{-1}a) using associativity and your hypothesis.
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  3. #3
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    Unhappy

    I'm sorry I don't think I understand. I could regroup by the associative property stating
    ab=(aba-1)a
    and I previously found that c=(aba-1) so by replacement ab=ca so therefore b=c. But I'm still confused about how to prove the group is abelian. I'm missing something.
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  4. #4
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    Well, you have ab=(aba^{-1})a

    You can define c:=aba^{-1} if you want, so, by hypothesis, you have: b=c, i.e. b=aba^{-1}, and you can conclude.
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  5. #5
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    yea i realized that a couple minutes after i had replied. thank you sooo much for the explanation!
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