Let $H \le G$, and $a \in G$ . Let $aHa^{-1} = \{aha^{-1} |h \in H \}$ . Show that $aHa^{-1}$ is a subgroup of $G$.

Suppose that $H \le G$ such that whenever $Ha \neq Hb$ then $aH \neq bH$ . Prove that $gHg^{-1} \subseteq H$ for all $g \in G$ .

If $H$ and $K$ are subgroups of orders $p$ and $n$ , respectively, where $p$ is a prime number, show that either $H \cap K=\{e\}$ or $H \le K$ .

If $G$ is a group and $K \le H \le G$ , then show that $[G:K] =[G:H][H:K]$ (assume finite indices). Hence deduce that if $[G:H] =p$, where $p$ is a prime, then either $H=K$ or $H=G$ .

If $G$ is a group and $H,K$ are two subgroups of finite indices in $G$, prove that $H \cap K$ is of finite index in $G$. Can you find an upper bound for the index of $H \cap K$ in $G$?

If $H$ and $K$ are subgroups of finite indices of a group $G$ such that $[G:H]$ and $[G:K]$ are relatively prime, show that $G =HK$ .

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.