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Math Help - Abelian Factor Group

  1. #1
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    Abelian Factor Group

    Question: Let G be a group and N be a normal subgroup of G. Show that the factor group G/N is abelian iff aba^-1b^-1 is in N for all a,b in G.

    If G/N is abelian then Na*Nb = Nb * Na and Naa^-1bb^-1 is in N and so Naba^-1b^-1 is in N

    If aba^-1b^-1 is in N then Nb*Na = Nba = aba^-1b^-1ba which canceles to ab which I would like to say is Nab = Na*Nb.

    I feel like I'm missing something though.

    Thanks
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by temp31415 View Post
    Question: Let G be a group and N be a normal subgroup of G. Show that the factor group G/N is abelian iff aba^-1b^-1 is in N for all a,b in G.

    If G/N is abelian then Na*Nb = Nb * Na and Naa^-1bb^-1 is in N and so Naba^-1b^-1 is in N

    If aba^-1b^-1 is in N then Nb*Na = Nba = aba^-1b^-1ba which canceles to ab which I would like to say is Nab = Na*Nb.

    I feel like I'm missing something though.

    Thanks
    yes, that was right but you can simplify it..

    Let G be a group and N be a normal subgroup of G.
    Let a,b \in G

    G/N is abelian \Longleftrightarrow aNbN = (ab)N = (ba)N = bNaN (let's take the middle..)

    \Longleftrightarrow (ab)(ba)^{-1} \in N definition of equal cosets..

    \Longleftrightarrow aba^{-1}b^{-1} \in N.. QED.
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  3. #3
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    Quote Originally Posted by temp31415 View Post
    Question: Let G be a group and N be a normal subgroup of G. Show that the factor group G/N is abelian iff aba^-1b^-1 is in N for all a,b in G.

    If G/N is abelian then Na*Nb = Nb * Na and Naa^-1bb^-1 is in N and so Naba^-1b^-1 is in N

    If aba^-1b^-1 is in N then Nb*Na = Nba = aba^-1b^-1ba which canceles to ab which I would like to say is Nab = Na*Nb.

    I feel like I'm missing something though.

    Thanks
    Try to prove a more useful related result.

    Definition:An element expressable as aba^{-1}b^{-1} is called a 'commutatator'.

    Definition:The 'commutatator subgroup' C is generated by all commutatators.

    Definition:The factor group G/N is abelian if and only if N contains the commutatator subgroup.
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