# Subgroup of cyclic normal subgroup is normal

• Apr 25th 2010, 12:49 PM
HomieG
Subgroup of cyclic normal subgroup is normal
Let $\displaystyle K \triangleleft G$ where $\displaystyle K$ is cyclic. Show that every subgroup of $\displaystyle K$ is normal in $\displaystyle G$.

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 $\displaystyle gxg^{-1} \in H \; x\in H, \forall g \in G$, where $\displaystyle x=h^i, \; i \in \mathbb{Z}$ with h the generator of the subgroup of K. Totally stuck though on where to go after this.
• Apr 25th 2010, 06:13 PM
tonio
Quote:

Originally Posted by HomieG
Let $\displaystyle K \triangleleft G$ where $\displaystyle K$ is cyclic. Show that every subgroup of $\displaystyle K$ is normal in $\displaystyle G$.

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 $\displaystyle gxg^{-1} \in H \; x\in H, \forall g \in G$, where $\displaystyle x=h^i, \; i \in \mathbb{Z}$ with h the generator of the subgroup of K. Totally stuck though on where to go after this.

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

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

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