Subgroup of cyclic normal subgroup is normal

Quote:

Originally Posted by HomieG

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