N asked in Science & MathematicsMathematics · 1 decade ago

Prove G/N is abelian if and only if aba^-1b^-1 in N for all a,b in G.?

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  • 1 decade ago
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    Suppose G/N is Abelian. Then,

    abN = (aN)(bN) = (bN)(aN) = baN

    So, N = (ab)(ba)^-1N = aba^-1b^-1N, which means aba^-1b^-1 is in N.

    Conversely, suppose aba^-1b^-1 is in N. Then,

    (bN)(aN)

    = baN

    = Nba

    =Naba^-1b^-1ba

    =Nab

    =abN

    =(aN)(bN)

    Thus, G/N is Abelian.

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