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Exercises 4.5 Exercises

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. \(\displaystyle 72 \in {\mathbb Z}_{240}\)
  6. \(\displaystyle 312 \in {\mathbb Z}_{471}\)
Hint.
(a) \(12\text{;}\) (c) infinite; (e) \(10\text{.}\)

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

11.

If \(a^{24} =e\) in a group \(G\text{,}\) what are the possible orders of \(a\text{?}\)
Hint.
\(1, 2, 3, 4, 6, 8, 12, 24\text{.}\)

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. \(\displaystyle (1+i)+\overline{(1+i)}\)
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) \(292\text{;}\) (c) \(1523\text{.}\)

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 \(2\text{,}\) 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.
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)]\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.
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