If every cyclic subgroup of a group G be normal in G, prove that every subgroup of G is normal in G.

Question

If every cyclic subgroup of a group G be normal in G, prove that every subgroup of G is normal in G.

Attepmt

Let G be a group. Let H be a Normal subgroup of G. Let $K=\langle a \rangle$ be a cyclic subgroup of G generated by $a\in G$. We shall have to show that H is normal in G. That is to prove that for $h\in H$, $ghg^{-1}\in G$ for all $g\in G$.

If the steps I have written, please help me to solve the problem.

Answer 1

Let $H$ be a subgroup of $G$, $h \in H$ and $g \in G$. You have to show that $g^ {-1}hg \in H$. Since $H$ is a subgroup, the cyclic group $\langle h \rangle$ generated by $h$, is a subgroup of $H$ and hence of $G$. Your premise tells us that $\langle h \rangle$ is normal, so $g^ {-1}hg \in \langle h \rangle$, since of course $h \in \langle h \rangle$. Hence $g^ {-1}hg=h^i$ for some $i$, and $h^i \in H$.