# Section8.4Exercises

##### 1

Let $G$ be a group and let $H\leq G$ have index 2. Prove that $H\unlhd G\text{.}$

##### 2

Let $G$ be an abelian group with $N\unlhd G\text{.}$ Prove that $G/N$ is abelian.

##### 3

Find the following.

1. $|2\Z/6\Z|$
2. $|H|\text{,}$ for $H=2+\langle 6\rangle \subseteq \Z_{12}$
3. $o(2+\langle 6\rangle)$ in $\Z_{12}/\langle 6\rangle$
4. $\langle fH\rangle$ in $D_4/H\text{,}$ where $H=\{e,r^2\}$
5. $|(\Z_6\times \Z_8)/(\langle 3\rangle\times \langle 2\rangle)|$
6. $|(\Z_{15} \times \Z_{24})/\langle (5,4)\rangle|$
##### 4

For each of the following, find a familiar group to which the given group is isomorphic. (Hint: Consider the group order, properties such as abelianness and cyclicity, group tables, orders of elements, etc.)

1. $\Z/14\Z$
2. $3\Z/12\Z$
3. $S_8/A_8$
4. $(\Z_4 \times \Z_{15})/(\langle 2 \rangle \times \langle 3 \rangle )$
5. $D_4/\langle r^2 \rangle$
##### 5

Let $H\unlhd G$ with index $k\text{,}$ and let $a\in G\text{.}$ Prove that $a^k\in H\text{.}$