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Math Help - Subgroup of cyclic normal subgroup is normal

  1. #1
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    Subgroup of cyclic normal subgroup is normal

    Let K \triangleleft G where K is cyclic. Show that every subgroup of K is normal in G.


    Been working on this for awhile and i know that every subgroup of a cyclic subgroup will be cyclic and correspondingly abelian and that we must show gxg^{-1} \in H \; x\in H, \forall g \in G, where x=h^i, \; i \in \mathbb{Z} with h the generator of the subgroup of K. Totally stuck though on where to go after this.
    Last edited by HomieG; April 25th 2010 at 03:33 PM.
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  2. #2
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    Quote Originally Posted by HomieG View Post
    Let K \triangleleft G where K is cyclic. Show that every subgroup of K is normal in G.


    Been working on this for awhile and i know that every subgroup of a cyclic subgroup will be cyclic and correspondingly abelian and that we must show gxg^{-1} \in H \; x\in H, \forall g \in G, where x=h^i, \; i \in \mathbb{Z} with h the generator of the subgroup of K. Totally stuck though on where to go after this.

    Hint: (1) Definition: a subgroup H of a group G is called characteristic if \phi(H)=H\,\,,\,\forall \phi\in Aut(G) , and we write this as H\, char.\, G

    Theorem: If K\triangleleft G\,\,\,and\,\,\,H\,char.\,K\,\,\,then\,\,\,H\trian  gleleft G

    Lemma: Any subgroup of a finite cyclic group is characteristic (hint: there exists only one of order any divisor of the group's order).

    Solve your problem now.

    Tonio
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