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Math Help - factor group

  1. #1
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    factor group

    Let G be any group, and N \triangleleft G. Show that aba^{-1}b^{-1}\in N, \forall a,b\in N \Longleftrightarrow G/N is abelian
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by GTK X Hunter View Post
    Let G be any group, and N \triangleleft G. Show that aba^{-1}b^{-1}\in N, \forall a,b\in N \Longleftrightarrow G/N is abelian
    Hint: Prove that aba^{-1}b^{-1}=1 \forall a,b \in G \Leftrightarrow G is abelian.
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  3. #3
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    Quote Originally Posted by Swlabr View Post
    Hint: Prove that aba^{-1}b^{-1}=1 \forall a,b \in G \Leftrightarrow G is abelian.
    do you mean aba^{-1}b^{-1}=e
    what is the reason?
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by GTK X Hunter View Post
    do you mean aba^{-1}b^{-1}=e
    what is the reason?
    1 and e are both commonly used to denote the identity element.

    What does this equality tell you? By definition, ab=ba \: \forall a, b \in G \Leftrightarrow G is abelian. Can you get this from the equation I gave you above?
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