## 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.

Hint.

(a) False; (c) false; (e) true.

### 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. 72 in $${\mathbb Z}_{240}$$

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

Hint.

(a) 12; (c) infinite; (e) 10.

### 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 $$(1 + \sqrt{3}\, i) / 2$$

Hint.

(a) $$7 {\mathbb Z} = \{ \ldots, -7, 0, 7, 14, \ldots \}\text{;}$$ (b) $$\{ 0, 3, 6, 9, 12, 15, 18, 21 \}\text{;}$$ (c) $$\{ 0 \}\text{,}$$ $$\{ 0, 6 \}\text{,}$$ $$\{ 0, 4, 8 \}\text{,}$$ $$\{ 0, 3, 6, 9 \}\text{,}$$ $$\{ 0, 2, 4, 6, 8, 10 \}\text{;}$$ (g) $$\{ 1, 3, 7, 9 \}\text{;}$$ (j) $$\{ 1, -1, i, -i \}\text{.}$$

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

Hint.

(a)

\begin{equation*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}. \end{equation*}

(c)

\begin{equation*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} -1 & 1 \\ -1 & 0 \end{pmatrix}, \\ \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}. \end{equation*}

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

Hint.

(a) $$0, 1, -1\text{;}$$ (b) $$1, -1$$

### 11.

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

Hint.

1, 2, 3, 4, 6, 8, 12, 24.

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

Hint.

(a) $$-3 + 3i\text{;}$$ (c) $$43- 18i\text{;}$$ (e) $$i$$

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

Hint.

(a) $$\sqrt{3} + i\text{;}$$ (c) $$-3\text{.}$$

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

Hint.

(a) $$\sqrt{2} \cis( 7 \pi /4)\text{;}$$ (c) $$2 \sqrt{2} \cis( \pi /4)\text{;}$$ (e) $$3 \cis(3 \pi/2)\text{.}$$

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

Hint.

(a) $$(1 - i)/2\text{;}$$ (c) $$16(i - \sqrt{3}\, )\text{;}$$ (e) $$-1/4\text{.}$$

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

Hint.

(a) 292; (c) 1523.

### 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 $$\langle g \rangle \cap \langle h \rangle \text{?}$$

Hint.

$$|\langle g \rangle \cap \langle h \rangle| = 1\text{.}$$

### 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 $$G\text{.}$$

Hint.

The identity element in any group has finite order. Let $$g, h \in G$$ have orders $$m$$ and $$n\text{,}$$ respectively. Since $$(g^{-1})^m = e$$ and $$(gh)^{mn} = e\text{,}$$ the elements of finite order in $$G$$ form a 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, show that $$G$$ must contain a subgroup of order 4. 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.

Hint.

If $$g$$ is an element distinct from the identity in $$G\text{,}$$ $$g$$ must generate $$G\text{;}$$ otherwise, $$\langle g \rangle$$ is a nontrivial proper subgroup of $$G\text{.}$$

### 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)]. \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.