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Math Help - Prove that G is abelian if and only if (ab)^(2) = a^(2) * b^(2)

  1. #1
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    Prove that G is abelian if and only if (ab)^(2) = a^(2) * b^(2)

    Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)^(2) = a^(2) * b^(2)
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  2. #2
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    What have you tried? Where are you getting stuck? (Also, I believe the statement should be that G is abelian  \Leftrightarrow (ab)^2 = a^2b^2 \ \forall a,b \ \in G)
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  3. #3
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    Quote Originally Posted by Defunkt View Post
    What have you tried? Where are you getting stuck? (Also, I believe the statement should be that G is abelian  \Leftrightarrow (ab)^2 = a^2b^2 \ \forall a,b \ \in G)

    i got stuck how to do it conversely.  \Leftrightarrow a^2b^2 = (ab)^2 \ \forall a,b \ \in G
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  4. #4
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    consider a,b \in G
    if a^2b^2 = (ab)^2 then aabb = abab
    since G is a group, a^{-1}, b^{-1} \in G
    consider a^{-1}(aabb)b^{-1} = a^{-1}(abab)b^{-1} and use the properties of groups and you'll see why G is abelian.
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