## Exercises 4.5 Exercises

### 1.

Prove or disprove each of the following statements.

All of the generators of \({\mathbb Z}_{60}\) are prime.

\(U(8)\) is cyclic.

\({\mathbb Q}\) is cyclic.

If every proper subgroup of a group \(G\) is cyclic, then \(G\) is a cyclic group.

A group with a finite number of subgroups is finite.

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

### 2.

Find the order of each of the following elements.

\(\displaystyle 5 \in {\mathbb Z}_{12}\)

\(\displaystyle \sqrt{3} \in {\mathbb R}\)

\(\displaystyle \sqrt{3} \in {\mathbb R}^\ast\)

\(\displaystyle -i \in {\mathbb C}^\ast\)

72 in \({\mathbb Z}_{240}\)

312 in \({\mathbb Z}_{471}\)

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

### 3.

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

The subgroup of \({\mathbb Z}\) generated by 7

The subgroup of \({\mathbb Z}_{24}\) generated by 15

All subgroups of \({\mathbb Z}_{12}\)

All subgroups of \({\mathbb Z}_{60}\)

All subgroups of \({\mathbb Z}_{13}\)

All subgroups of \({\mathbb Z}_{48}\)

The subgroup generated by 3 in \(U(20)\)

The subgroup generated by 5 in \(U(18)\)

The subgroup of \({\mathbb R}^\ast\) generated by 7

The subgroup of \({\mathbb C}^\ast\) generated by \(i\) where \(i^2 = -1\)

The subgroup of \({\mathbb C}^\ast\) generated by \(2i\)

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

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

(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.

\(\displaystyle \displaystyle \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\)

\(\displaystyle \displaystyle \begin{pmatrix} 0 & 1/3 \\ 3 & 0 \end{pmatrix}\)

\(\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}\)

\(\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}\)

\(\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ -1 & 0 \end{pmatrix}\)

\(\displaystyle \displaystyle \begin{pmatrix} \sqrt{3}/ 2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}\)

(a)

(c)

### 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.

\(\displaystyle {\mathbb Z}\)

\(\displaystyle {\mathbb Q}^\ast\)

\(\displaystyle {\mathbb R}^\ast\)

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

### 11.

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

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

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.

\(\displaystyle (3-2i)+ (5i-6)\)

\(\displaystyle (4-5i)-\overline{(4i -4)}\)

\(\displaystyle (5-4i)(7+2i)\)

\(\displaystyle (9-i) \overline{(9-i)}\)

\(\displaystyle i^{45}\)

\(\displaystyle (1+i)+\overline{(1+i)}\)

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

### 16.

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

\(\displaystyle 2 \cis(\pi / 6 )\)

\(\displaystyle 5 \cis(9\pi/4)\)

\(\displaystyle 3 \cis(\pi)\)

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

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

### 17.

Change the following complex numbers to polar representation.

\(\displaystyle 1-i\)

\(\displaystyle -5\)

\(\displaystyle 2+2i\)

\(\displaystyle \sqrt{3} + i\)

\(\displaystyle -3i\)

\(\displaystyle 2i + 2 \sqrt{3}\)

(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.

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

\(\displaystyle (1 - i)^{6}\)

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

\(\displaystyle (-i)^{10}\)

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

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

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

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

### 19.

Prove each of the following statements.

\(\displaystyle |z| = | \overline{z}|\)

\(\displaystyle z \overline{z} = |z|^2\)

\(\displaystyle z^{-1} = \overline{z} / |z|^2\)

\(\displaystyle |z +w| \leq |z| + |w|\)

\(\displaystyle |z - w| \geq | |z| - |w||\)

\(\displaystyle |z w| = |z| |w|\)

### 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.

\(\displaystyle 292^{3171} \pmod{ 582}\)

\(\displaystyle 2557^{ 341} \pmod{ 5681}\)

\(\displaystyle 2071^{ 9521} \pmod{ 4724}\)

\(\displaystyle 971^{ 321} \pmod{ 765}\)

(a) 292; (c) 1523.

### 23.

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

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

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

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 \(\langle g \rangle \cap \langle h \rangle \text{?}\)

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

### 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 \(G\text{.}\)

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.

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

### 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.