Show that if every cyclic subgroup of a group $G$ is normal in $G$, then every subgroup of $G$ is normal.

Suppose that $H$ is the only subgroup of order $\mid H \mid$ in a finite group $G$. Prove that $H$ char $G$, and hence  $H \unlhd G$ .

Let $H \le G \ and \ N(H) = \{g\in G \mid gHg^{-1}=H \}$. Prove that

(i) $N(H) \le G$,    (ii) $H \unlhd N(H)$,    (iii) If $K \le G$ such that $H \unlhd K$, then $K \subseteq N(H)$,

(iv) $H \unlhd G$ iff $N(H)=G$.

If $G$ is a finite group and $N \unlhd G$ such that $\mid N \mid$ and $[G:N]$ are coprime, show that for any $a \in G ,a ^{\mid N \mid } = e \Rightarrow a \in N$.

If $H \le G$, such that $x^2 \in H$ for all $x \in G$. Prove that $H \unlhd G$ and $G/H$ is abelian.

Show that if $N$ is a normal subgroup of finite index in a group $G$, and $H$ is a subgroup of finite order in $G$ such that $[G:N]$ and $\mid H \mid$ are relatively prime, then $H \subseteq N$.

Show that if $N$ is a normal subgroup of finite order in a group $G$, and $H$ is a subgroup of finite index in $G$ such that $[G:H]$ and $\mid N \mid$ are relatively prime, then $N \subseteq H$.

Show that if $G$ is a group with centre $Z(G)$, and if $G/Z(G)$ is cyclic, then $G$ is abelian. Hence show that there does not exist any group $G$ such that $\mid G/Z(G) \mid = p$, where $p$ is a prime number.

Prove that $\mathbb{Q/Z}$ is isomorphic to the multiplicative group of root of unity in $\mathbb{C^\times}$.

Show that $(\mathbb{Q},+)$ has no proper subgroup of finite index. Deduce that $\mathbb{Q/Z}$ has no proper subgroups of finite index.

Show that an infinite group is cyclic if and only if it is isomorphic to each of its nontrivial subgroups.