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## Exercises4.5Exercises

###### 1.

Prove or disprove each of the following statements.

1. All of the generators of ${\mathbb Z}_{60}$ are prime.

2. $U(8)$ is cyclic.

3. ${\mathbb Q}$ is cyclic.

4. If every proper subgroup of a group $G$ is cyclic, then $G$ is a cyclic group.

5. A group with a finite number of subgroups is finite.

###### 2.

Find the order of each of the following elements.

1. $\displaystyle 5 \in {\mathbb Z}_{12}$

2. $\displaystyle \sqrt{3} \in {\mathbb R}$

3. $\displaystyle \sqrt{3} \in {\mathbb R}^\ast$

4. $\displaystyle -i \in {\mathbb C}^\ast$

5. $\displaystyle 72 \in {\mathbb Z}_{240}$

6. $\displaystyle 312 \in {\mathbb Z}_{471}$

###### 3.

List all of the elements in each of the following subgroups.

1. The subgroup of ${\mathbb Z}$ generated by $7$

2. The subgroup of ${\mathbb Z}_{24}$ generated by $15$

3. All subgroups of ${\mathbb Z}_{12}$

4. All subgroups of ${\mathbb Z}_{60}$

5. All subgroups of ${\mathbb Z}_{13}$

6. All subgroups of ${\mathbb Z}_{48}$

7. The subgroup generated by 3 in $U(20)$

8. The subgroup generated by 5 in $U(18)$

9. The subgroup of ${\mathbb R}^\ast$ generated by $7$

10. The subgroup of ${\mathbb C}^\ast$ generated by $i$ where $i^2 = -1$

11. The subgroup of ${\mathbb C}^\ast$ generated by $2i$

12. The subgroup of ${\mathbb C}^\ast$ generated by $(1 + i) / \sqrt{2}$

13. The subgroup of ${\mathbb C}^\ast$ generated by $(1 + \sqrt{3}\, i) / 2$

###### 4.

Find the subgroups of $GL_2( {\mathbb R })$ generated by each of the following matrices.

1. $\displaystyle \displaystyle \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

2. $\displaystyle \displaystyle \begin{pmatrix} 0 & 1/3 \\ 3 & 0 \end{pmatrix}$

3. $\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}$

4. $\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}$

5. $\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ -1 & 0 \end{pmatrix}$

6. $\displaystyle \displaystyle \begin{pmatrix} \sqrt{3}/ 2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}$

###### 5.

Find the order of every element in ${\mathbb Z}_{18}\text{.}$

###### 6.

Find the order of every element in the symmetry group of the square, $D_4\text{.}$

###### 7.

What are all of the cyclic subgroups of the quaternion group, $Q_8\text{?}$

###### 8.

List all of the cyclic subgroups of $U(30)\text{.}$

###### 9.

List every generator of each subgroup of order 8 in ${\mathbb Z}_{32}\text{.}$

###### 10.

Find all elements of finite order in each of the following groups. Here the “$\ast$” indicates the set with zero removed.

1. $\displaystyle {\mathbb Z}$

2. $\displaystyle {\mathbb Q}^\ast$

3. $\displaystyle {\mathbb R}^\ast$

###### 11.

If $a^{24} =e$ in a group $G\text{,}$ what are the possible orders of $a\text{?}$

###### 12.

Find a cyclic group with exactly one generator. Can you find cyclic groups with exactly two generators? Four generators? How about $n$ generators?

###### 13.

For $n \leq 20\text{,}$ which groups $U(n)$ are cyclic? Make a conjecture as to what is true in general. Can you prove your conjecture?

###### 14.

Let

\begin{equation*} A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \qquad \text{and} \qquad B = \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix} \end{equation*}

be elements in $GL_2( {\mathbb R} )\text{.}$ Show that $A$ and $B$ have finite orders but $AB$ does not.

###### 15.

Evaluate each of the following.

1. $\displaystyle (3-2i)+ (5i-6)$

2. $\displaystyle (4-5i)-\overline{(4i -4)}$

3. $\displaystyle (5-4i)(7+2i)$

4. $\displaystyle (9-i) \overline{(9-i)}$

5. $\displaystyle i^{45}$

6. $\displaystyle (1+i)+\overline{(1+i)}$

###### 16.

Convert the following complex numbers to the form $a + bi\text{.}$

1. $\displaystyle 2 \cis(\pi / 6 )$

2. $\displaystyle 5 \cis(9\pi/4)$

3. $\displaystyle 3 \cis(\pi)$

4. $\displaystyle \cis(7\pi/4) /2$

###### 17.

Change the following complex numbers to polar representation.

1. $\displaystyle 1-i$

2. $\displaystyle -5$

3. $\displaystyle 2+2i$

4. $\displaystyle \sqrt{3} + i$

5. $\displaystyle -3i$

6. $\displaystyle 2i + 2 \sqrt{3}$

###### 18.

Calculate each of the following expressions.

1. $\displaystyle (1+i)^{-1}$

2. $\displaystyle (1 - i)^{6}$

3. $\displaystyle (\sqrt{3} + i)^{5}$

4. $\displaystyle (-i)^{10}$

5. $\displaystyle ((1-i)/2)^{4}$

6. $\displaystyle (-\sqrt{2} - \sqrt{2}\, i)^{12}$

7. $\displaystyle (-2 + 2i)^{-5}$

###### 19.

Prove each of the following statements.

1. $\displaystyle |z| = | \overline{z}|$

2. $\displaystyle z \overline{z} = |z|^2$

3. $\displaystyle z^{-1} = \overline{z} / |z|^2$

4. $\displaystyle |z +w| \leq |z| + |w|$

5. $\displaystyle |z - w| \geq | |z| - |w||$

6. $\displaystyle |z w| = |z| |w|$

###### 20.

List and graph the 6th roots of unity. What are the generators of this group? What are the primitive 6th roots of unity?

###### 21.

List and graph the 5th roots of unity. What are the generators of this group? What are the primitive 5th roots of unity?

###### 22.

Calculate each of the following.

1. $\displaystyle 292^{3171} \pmod{ 582}$

2. $\displaystyle 2557^{ 341} \pmod{ 5681}$

3. $\displaystyle 2071^{ 9521} \pmod{ 4724}$

4. $\displaystyle 971^{ 321} \pmod{ 765}$

###### 23.

Let $a, b \in G\text{.}$ Prove the following statements.

1. The order of $a$ is the same as the order of $a^{-1}\text{.}$

2. For all $g \in G\text{,}$ $|a| = |g^{-1}ag|\text{.}$

3. The order of $ab$ is the same as the order of $ba\text{.}$

###### 24.

Let $p$ and $q$ be distinct primes. How many generators does ${\mathbb Z}_{pq}$ have?

###### 25.

Let $p$ be prime and $r$ be a positive integer. How many generators does ${\mathbb Z}_{p^r}$ have?

###### 26.

Prove that ${\mathbb Z}_{p}$ has no nontrivial subgroups if $p$ is prime.

###### 27.

If $g$ and $h$ have orders $15$ and $16$ respectively in a group $G\text{,}$ what is the order of $\langle g \rangle \cap \langle h \rangle \text{?}$

###### 28.

Let $a$ be an element in a group $G\text{.}$ What is a generator for the subgroup $\langle a^m \rangle \cap \langle a^n \rangle\text{?}$

###### 29.

Prove that ${\mathbb Z}_n$ has an even number of generators for $n \gt 2\text{.}$

###### 30.

Suppose that $G$ is a group and let $a\text{,}$ $b \in G\text{.}$ Prove that if $|a| = m$ and $|b| = n$ with $\gcd(m,n) = 1\text{,}$ then $\langle a \rangle \cap \langle b \rangle = \{ e \}\text{.}$

###### 31.

Let $G$ be an abelian group. Show that the elements of finite order in $G$ form a subgroup. This subgroup is called the torsion subgroup of $G\text{.}$

###### 32.

Let $G$ be a finite cyclic group of order $n$ generated by $x\text{.}$ Show that if $y = x^k$ where $\gcd(k,n) = 1\text{,}$ then $y$ must be a generator of $G\text{.}$

###### 33.

If $G$ is an abelian group that contains a pair of cyclic subgroups of order $2\text{,}$ show that $G$ must contain a subgroup of order $4\text{.}$ Does this subgroup have to be cyclic?

###### 34.

Let $G$ be an abelian group of order $pq$ where $\gcd(p,q) = 1\text{.}$ If $G$ contains elements $a$ and $b$ of order $p$ and $q$ respectively, then show that $G$ is cyclic.

###### 35.

Prove that the subgroups of $\mathbb Z$ are exactly $n{\mathbb Z}$ for $n = 0, 1, 2, \ldots\text{.}$

###### 36.

Prove that the generators of ${\mathbb Z}_n$ are the integers $r$ such that $1 \leq r \lt n$ and $\gcd(r,n) = 1\text{.}$

###### 37.

Prove that if $G$ has no proper nontrivial subgroups, then $G$ is a cyclic group.

###### 38.

Prove that the order of an element in a cyclic group $G$ must divide the order of the group.

###### 39.

Prove that if $G$ is a cyclic group of order $m$ and $d \mid m\text{,}$ then $G$ must have a subgroup of order $d\text{.}$

###### 40.

For what integers $n$ is $-1$ an $n$th root of unity?

###### 41.

If $z = r( \cos \theta + i \sin \theta)$ and $w = s(\cos \phi + i \sin \phi)$ are two nonzero complex numbers, show that

\begin{equation*} zw = rs[ \cos( \theta + \phi) + i \sin( \theta + \phi)]\text{.} \end{equation*}
###### 42.

Prove that the circle group is a subgroup of ${\mathbb C}^*\text{.}$

###### 43.

Prove that the $n$th roots of unity form a cyclic subgroup of ${\mathbb T}$ of order $n\text{.}$

###### 44.

Let $\alpha \in \mathbb T\text{.}$ Prove that $\alpha^m =1$ and $\alpha^n = 1$ if and only if $\alpha^d = 1$ for $d = \gcd(m,n)\text{.}$

###### 45.

Let $z \in {\mathbb C}^\ast\text{.}$ If $|z| \neq 1\text{,}$ prove that the order of $z$ is infinite.

###### 46.

Let $z =\cos \theta + i \sin \theta$ be in ${\mathbb T}$ where $\theta \in {\mathbb Q}\text{.}$ Prove that the order of $z$ is infinite.