Exercises

1. Prove that conjugacy is an equivalence relation on the set of all subgroups of a group. Describe the equivalence classes.

2. Let $G$ be a group and $H \leq G$. Prove

   (1) $H \subset N(H)$,

   (2) $N(H) \leq G$.

3. Prove Theorem 6.1.1.

4. Prove that any subgroup of index $2$ is a normal subgroup.

   (HINT: Consider the partition in terms of left cosets and then in terms of right cosets.)

5. Let $G$ be a group and $H \leq G$.

   (a) Prove that $aHa^{-1}\subset H$ for all $a \in G$ implies $aHa^{-1} = H$.

   (b) Suppose $H$ has the property that the product of any two left cosets of $H$ is also a left coset of $H$; then prove that $H \lhd G$.

   (HINT: For (a) note that the condition must be true for $a^{- 1}$. For (b) note that the product of $aH$ and $a^{-1}H$ must be a left coset. Moreover $e \in aHa^{-1}H$. Then apply (a).)

6. Show that $H =\{ 1, (1,2)\} = \langle (1,2)\rangle$ is not a normal subgroup of $S_3$.

7. Referring to the Cayley table for $A_4$, let
   
   $$V_4 = \{(1), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)\},$$
   
   
   i.e., $V_4 = \{f_1, f_2, f_3, f_4\}$.

   (a)Prove $V_4 \leq A_4$.

   (b) Write all the left cosets of $V_4$ in $A_4$ and then write all the right cosets of $V_4$ in $A_4$.

   (c) Use the result of (b) to show that $V_4 \lhd A_4$.

8. Let $G$ be a group, $N \lhd G$, and $H \lhd N$. Does it follow that $H \lhd G$ (i.e., is the relation of being a normal subgroup transitive)?

   (HINT: Think of $A_4$, $V_4$, and $\langle (1,2)(3,4)\rangle$.)

9. Let $G$ be a group and $N \leq G$. Prove $N \lhd G$ if and only if $N$ consists of complete conjugacy classes of all its elements, i.e., $Cl(g) \subset N$ for all $g \in N$.