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    Description of Abelian group

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    Modern Algebra
    Group Theory (VII)

    To prove that if G is an abelian group, then for all a,b belongs to G and all integers n, (a.b)^n=a^n.b^n.

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    Prove that if is an abelian group, then for all and all integers , .

    Proof:- Let be arbitrary.
    By hypothesis, is an abelian group,

    which imply that ---------------------------------(1)

    which imply that


    Suppose where is any integer, then


    Hence by mathematical induction,

    for all ...

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    This shows how to complete a proof regarding an Abelian group. The solution is detailed and well presented.