Let $G$ be a finite group, and suppose that for any two subgroups $H$ and $K$ of $G$ either $H \subseteq K \ or \ K \subseteq H$. Prove that $G$ is a cyclic group of prime power order.

Show that the direct product of two infinite cyclic groups in not cyclic.

Determine the number of subgroups of $(\mathbb{Z_{360}},+)$. Also determine the number of automorphisms of $(\mathbb{Z_{360}},+)$.

Prove that the homomorphic image of a cyclic (abelian) group is cyclic (abelian). Give an example to show that if the homomorphic image of a group is cyclic (abelian), the same may not hold for the group itself.

In $(\mathbb{Z_{24}},+)$ find all $m \in \mathbb{Z_{24}}$ such that $\langle 21 \rangle = \langle m \rangle$.

Show that $(\mathbb{Q^+}, .)$, the multiplicative group of positive rational numbers, is not a cyclic group.

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.