Let $f:G \rightarrow G^\prime$ be a homomorphism of groups. Show that if $H^\prime \unlhd G^\prime$ , then $f^{-1}(H^\prime ) \unlhd G$ and $\ker f \subseteq f^{-1}(H^\prime)$. Further show that if $f$ is surjective and $H \unlhd G$, then $f(H) \unlhd G^\prime$ .

Show by an example that we can find groups $G,H,K$ such that $K \unlhd G$ and $H \unlhd K$, but $H$ is not a normal subgroup of $G$.

Give an example of a group $G$ and a subgroup $H$ of $G$ such that for some $a \in G , aHa^{-1} \subsetneq H$.

If a cyclic subgroup $T$ of a group $G$ is normal in $G$, then show that every subgroup of $T$ is normal in $G$.

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 $N \subseteq H$.