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# Section7.4Exercises

##### 1

How many distinct partitions of the set $S=\{a,b,c,d\}$ are there? You do not need to list them. (Yes, you can find this answer online. But I recommend doing the work yourself for practice working with partitions!)

##### 2

1. Let $n\in \Z^+\text{.}$ Prove that $\equiv_n$ is an equivalence relation on $\Z\text{.}$

2. The cells of the induced partition of $\Z$ are called the residue classes (or congruence classes) of $\Z$ modulo $n$. Using set notation of the form $\{\ldots,\#, \#,\#,\ldots\}$ for each class, write down the residue classes of $\Z$ modulo $4\text{.}$

##### 3

Let $G$ be a group with subgroup $H\text{.}$ Prove that $\simr$ is an equivalence relation on $G\text{.}$

##### 4

Find the indices of:

1. $H=\langle (15)(24)\rangle$ in $S_5$

2. $K=\langle (2354)(34)\rangle$ in $S_6$

3. $A_n$ in $S_n$

##### 5

For each subgroup $H$ of group $G\text{,}$ (i) find the left and the right cosets of $H$ in $G\text{,}$ (ii) decide whether or not $H$ is normal in $G\text{,}$ and (iii) find $(G:H)\text{.}$

Write all permutations using disjoint cycle notation, and write all dihedral group elements using standard form.

1. $H=6\Z$ in $G=2\Z$

2. $H=\langle 4\rangle$ in $\Z_{20}$

3. $H=\langle (23)\rangle$ in $G=S_3$

4. $H=\langle r\rangle$ in $G=D_4$

5. $H=\langle f\rangle$ in $G=D_4$

##### 6

For each of the following, give an example of a group $G$ with a subgroup $H$ that matches the given conditions. If no such example exists, prove that.

1. A group $G$ with subgroup $H$ such that $|G/H|=1\text{.}$

2. A finite group $G$ with subgroup $H$ such that $|G/H|=|G|\text{.}$

3. An abelian group $G$ of order $8$ containing a non-normal subgroup $H$ of order 2.

4. A group $G$ of order 8 containing a normal subgroup of order $2\text{.}$

5. A nonabelian group $G$ of order 8 containing a normal subgroup of index $2\text{.}$

6. A group $G$ of order 8 containing a subgroup of order $3\text{.}$

7. An infinite group $G$ containing a subgroup $H$ of finite index.

8. An infinite group $G$ containing a finite nontrivial subgroup $H\text{.}$

##### 7

True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem. Throughout, let $G$ be a group with subgroup $H$ and elements $a,b\in G\text{.}$

1. If $a\in bH$ then $aH$ must equal $bH\text{.}$

2. $aH$ must equal $Ha\text{.}$

3. If $aH=bH$ then $Ha$ must equal $Hb\text{.}$

4. If $a\in H$ then $aH$ must equal $Ha\text{.}$

5. $H$ must be normal in $G$ if there exists $a\in G$ such that $aH=Ha\text{.}$

6. If $aH=bH$ then $ah=bh$ for every $h\in H\text{.}$

7. If $G$ is finite, then $|G/H|$ must be less than $|G|\text{.}$

8. If $G$ is finite, then $(G:H)$ must be less than or equal to $|G|\text{.}$

##### 8

Let $G$ be a group of order $pq\text{,}$ where $p$ and $q$ are prime, and let $H$ be a proper subgroup of $G\text{.}$ Prove that $H$ is cyclic.

##### 9

Prove Corollary 7.3.10: that is, let $G$ be a group of prime order, and prove that $G$ is cyclic.