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Section4.4Exercises

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

  2. \(\sqrt{3} \in {\mathbb R}\)

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

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

  5. \(72 \in {\mathbb Z}_{240}\)

  6. \(312 \in {\mathbb Z}_{471}\)

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

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

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

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

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

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

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 β€œ\(\ast\)” indicates the set with zero removed.

  1. \({\mathbb Z}\)

  2. \({\mathbb Q}^\ast\)

  3. \({\mathbb R}^\ast\)

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

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

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

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

  5. \(i^{45}\)

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

16

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

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

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

  3. \(3 \cis(\pi)\)

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

17

Change the following complex numbers to polar representation.

  1. \(1-i\)

  2. \(-5\)

  3. \(2+2i\)

  4. \(\sqrt{3} + i\)

  5. \(-3i\)

  6. \(2i + 2 \sqrt{3}\)

18

Calculate each of the following expressions.

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

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

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

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

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

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

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

19

Prove each of the following statements.

  1. \(|z| = | \overline{z}|\)

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

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

  4. \(|z +w| \leq |z| + |w|\)

  5. \(|z - w| \geq | |z| - |w||\)

  6. \(|z w| = |z| |w|\)

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

  2. \(2557^{ 341} \pmod{ 5681}\)

  3. \(2071^{ 9521} \pmod{ 4724}\)

  4. \(971^{ 321} \pmod{ 765}\)

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)]. \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.

This group of exercises is about the group of units mod \(n\text{,}\) \(U(n)\text{,}\) which is sometimes cyclic, sometimes not. There are some commands in Sage that will answer some of these questions very quickly, but instead of using those now, just use the basic techniques described. The idea here is to just work with elements, and lists of elements, to discern the subgroup structure of these groups.

Sage worksheets have extensive capabilities for making new cells with carefully formatted text, include support for syntax to express mathematics. So when a question asks for explanation or commentary, make a new cell and communicate clearly with your audience. Continue this practice in subsequent exercise sets.

47Sage Exercise 1

Execute the statement R = Integers(40) to create the set [0,1,2,...,39] This is a group under addition mod \(40\text{,}\) which we will ignore. Instead we are interested in the subset of elements which have an inverse under multiplication mod \(40\text{.}\) Determine how big this subgroup is by executing the command R.unit_group_order(), and then obtain a list of these elements with R.list_of_elements_of_multiplicative_group().

48Sage Exercise 2

You can create elements of this group by coercing regular integers into U, such as with the statement a = U(7). (Don't confuse this with our mathematical notation \(U(40)\text{.}\)) This will tell Sage that you want to view \(7\) as an element of \(U\text{,}\) subject to the corresponding operations. Determine the elements of the cyclic subgroup of \(U\) generated by \(7\) with a list comprehension as follows:

What is the order of \(7\) in \(U(40)\text{?}\)

49Sage Exercise 3

The group \(U(49)\) is cyclic. Using only the Sage commands described previously, use Sage to find a generator for this group. Now using only theorems about the structure of cyclic groups, describe each of the subgroups of \(U(49)\) by specifying its order and by giving an explicit generator. Do not repeat any of the subgroups β€” in other words, present each subgroup exactly once. You can use Sage to check your work on the subgroups, but your answer about the subgroups should rely only on theorems and be a nicely written paragraph with a table, etc.

50Sage Exercise 4

The group \(U(35)\) is not cyclic. Again, using only the Sage commands described previously, use computations to provide irrefutable evidence of this. How many of the \(16\) different subgroups of \(U(35)\) can you list?

51Sage Exercise 5

Again, using only the Sage commands described previously, explore the structure of \(U(n)\) for various values of \(n\) and see if you can formulate an interesting conjecture about some basic property of this group. (Yes, this is a very open-ended question, but this is ultimately the real power of exploring mathematics with Sage.)