Let $G$ be a finite group. Show that the only homomorphism from $G$ to $(\mathbb{Z} ,+)($ or to $(\mathbb{Q} ,+)$ or to $(\mathbb{R} ,+) )$ is the zero homomorphism.

Show that any endomorphism of $( \mathbb{Q} ,+)$ is multiplication by a rational number.

 Prove that there is no non-trivial homomorphism from

(i) $(\mathbb{Q} ,+)$ to $(\mathbb{Z} ,+)$

(ii) $(\mathbb{Q} ,+)$ to $(\mathbb{Z_m} ,+)$

(iii) $(\mathbb{Q} ,+)$to $(\mathbb{Q^\ast} ,\times)$, where $\mathbb{Q^\ast} = \mathbb{Q} - \{0\}$.

Prove that there is no surjective homomorphism from

(i) $\mathbb{Z_{16}} \times \mathbb{Z_{2}}$ to $\mathbb{Z_{4}} \times \mathbb{Z_{4}}$        (ii) $\mathbb{Z_{4}} \times \mathbb{Z_{4}}$ to $\mathbb{Z_8}$ .

If $\phi$ is a homomorphism from $\mathbb{Z_{30}}$ onto a group of order 5, determine ker $\phi$ .

Let $G$ be an abelian group and $n$ a positive integer. Define $f: G \rightarrow G$ by $f(x) =x^n$ . Show that $f$ is an endomorphism of $G$. Further if $G$ is finite such that $n$ and $|G|$ are coprime, show that $f$ is an automorphism of $G$ .

If a group $G$ contains an element a  with exactly two conjugates, then show that $G$ has a normal subgroup $N \neq \{e\}$ .

Let $H$ be a proper subgroup of a finite group $G$ . Show that $G$ contains at least one element that is not in any conjugate of $H$ .

Let $G$ be a non-abelian group of order 21. Prove that  $Z(G)=\{e\}$ . Hence show that there is at most one possible class equation for $G$.

Let $G$ be an abelian group and $H,K$ two finite subgroups of $G$ of orders $m$ and $n$, respectively. Prove that $G$ has a subgroup of order $LCM(m,n)$ .

(a) Show that the center of a group $G$ is a characteristic subgroup of $G$.

(b) Prove that characteristic subgroups are normal. Give an example of a normal subgroup which is not characteristic.

(c) Show that if $H$ is the only subgroup of order $n$ in $G$, or if $H$ is the only subgroup of index $k$ in $G$, then $H$ is characteristic in $G$.