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Math Help - Cyclic subgroups

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    Cyclic subgroups

    If X is a cyclic normal subgroup of Y I need to prove that any subgroup of X is also a normal subgroup of Y.

    This seems trivial but I'm not sure that it is.
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  2. #2
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    Quote Originally Posted by d_p_osters View Post
    If X is a cyclic normal subgroup of Y I need to prove that any subgroup of X is also a normal subgroup of Y.

    This seems trivial but I'm not sure that it is.
    If X has generator a, and Z is a subgroup of X, generated by a^k, then Z is the set of all elements of the form a^{nk}.

    If g\in G, then g^{-1}Zg is generated by g^{-1}a^kg = (g^{-1}ag)^k, from which you can see that g^{-1}ZG\subseteq Z.
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