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Math Help - quotient groups

  1. #1
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    quotient groups

    question: Give an examples of a group G and a normal subgroup N satisfying the following properties; or if no example exists, state that. In either case, justify answers.

    (a)G is abelian, and G/N is non-abelian
    (b)G is cyclic, and G/N is not cyclic
    (c)G/N is not isomorphic to any subgroup G.

    could anyone help me with this problem??? please?
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  2. #2
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    Quote Originally Posted by jin_nzzang View Post
    question: Give an examples of a group G and a normal subgroup N satisfying the following properties; or if no example exists, state that. In either case, justify answers.

    (a)G is abelian, and G/N is non-abelian
    (b)G is cyclic, and G/N is not cyclic
    (c)G/N is not isomorphic to any subgroup G.

    could anyone help me with this problem??? please?
    no example exists for (a) and (b). for (c) an example is G=\mathbb{Z} and N=2\mathbb{Z}.
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  3. #3
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    Hi
    thanks a lot for your help,, but could you please justify the answers so that i can understand it better? (at least one of them as an example ? )
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by jin_nzzang View Post
    Hi
    thanks a lot for your help,, but could you please justify the answers so that i can understand it better? (at least one of them as an example ? )
    Remember that your quotient group acts very much like your group - it has multiplication (Nx)(Ny)=N(xy).

    So, if the group is abelian then (Nx)(Ny)=N(xy) = N(yx) = (Ny)(Nx).
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