Exercises

1. Prove that if $H_1\leq G$, $H_2 \leq G$, $G$ a group, with $H_2 \lhd G$ then the map $\phi$ given by $\phi(h_1) = H_1H_2$, for $h_1 \in H_1$ is a homomorphism from $H_1$ into $H_1H_2/H_2$.

2. If $N \lhd G$, $G$ a finite group, and if $[G : N]$ and $\vert N\vert$ are relatively prime, then show that $N$ contains every subgroup of $G$ whose order is a divisor of $\vert N\vert$.

   (HINT: Let $H \leq G$ such that $\vert H\vert\, \vert\, \vert N\vert$. Let $h\in H$ and consider $o(h)$ and $o(gN)$, for $gN$ an element of $G/N$. Use this to prove $h \in N$.)

3. Let $G$ be a group, $H$ a subgroup of $G$, let $C$ be a class of conjugate elements, and let $h\in H$ be fixed. Prove: $\vert Ha \cap C\vert= \vert Hah \cap C\vert$.

   (HINT: Define a map and prove it is 1-1 and onto.)

4. Let $H_1\leq G$, $H_2 \leq G$, $G$ a group, such that $H_2 \lhd H_1$; also let $H$ be any subgroup of $G$. Let $H_3=H_2\cap H$ and $H_4=H_1\cap H$. Prove the following statements.

   (a) $H_3\lhd H_4$.

   (b) $H_4/H_3$ is isomorphic to a subgroup of $H_1/H_2$. (HINT: Use the second isomorphism theorem ``appropriately''.