Let G be a group. Every cyclic subgroup of G is normal, then prove that every subgroup of G is normal

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- Aug 3rd 2008, 05:10 AMmathemanyakNormal2
Let G be a group. Every cyclic subgroup of G is normal, then prove that every subgroup of G is normal

- Aug 3rd 2008, 07:29 AMJaneBennet
Let $\displaystyle H$ be a subgroup of $\displaystyle G$, and let $\displaystyle g\in G,\ h\in H$. Then $\displaystyle \left<h\right>$ is normal in $\displaystyle G$ and so $\displaystyle ghg^{-1}=h^n\in H$ for some $\displaystyle n\in\mathbb{Z}$. Hence $\displaystyle H\vartriangleleft G$