almost complete need a few pointers to complete

Discussion in 'General Math' started by Abbey Vanderah, May 9, 2004.

  1. I have this problem almost done but I'm not really sure how to finish
    it out. So if someone could please help me with this question it would
    greatly be appreciated.

    Question: Let G be a group. If |G|=30 and |Z(G)|=5, then determine the
    isomorphsim class of G/Z(G).

    Proof: Since G/Z(G) has order 30/5=6, there are two groups of order 6,
    the cyclic group Z6, and the dihedral group S3. I think that the
    isomorphism class of G/Z(G) is Z6 but I'm not really sure. Could
    someone please explain to me how to get the isomorphism class.

    Thanks for the help.
    Abbey Vanderah, May 9, 2004
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  2. If G / Z(G) were isomorphic to Z6, then G / Z(G) would be cyclic
    (since Z6 is cyclic).

    There is a theorem that says if G / Z(G) is cyclic, then G is abelian.
    This would imply that G = Z(G).

    However, you have specified that Z(G) has order 5 and so Z(G) cannot
    equal G. Hence, G is not abelian and G / Z(G) is not cyclic.

    It looks to me like you are forced into G / Z(G) being S3.

    Hope this helps,

    Brian VanPelt, May 9, 2004
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