For any subgroup $H$ of $G$ and any non-empty subset $A$ of $G$ define $N_H(A) = \{ h \in H | hAh^{-1} = A \}$ . Show that $N_H(A) =N_G(A) \cap H$ and dedue that $N_H(A) \le H$ .

Let $h \le G$ . Let $N = \bigcap_{x \in G} xHx^{-1}$ . Prove that $N \le G$, and $aNa^{-1} =N$ for all $a \in G$ .

 If $H$ is a subgroup of finite index in $G$ , prove that there is only a finite number of distinct subgroups in $G$ of the form $aHa^ {-1}$ .

 If $H$ is of finite index in $G$, prove that there is a subgroup $N$ of $G$, contained in $H$ and of finite index in $G$ such that $aNa^{-1}=N$ for all $a \in G$ .

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 $N \neq \{e\}$ .