<exercises xml:id="exercises-cyclic" xml:base="exercises/cyclic.xml">
<title>Exercises</title>
<exercise number="1">
<statement>
<p>
Prove or disprove each of the following statements.
</p>
<ol>
<li>
<p>
All of the generators of <m>{\mathbb Z}_{60}</m> are prime.
</p>
</li>
<li>
<p>
<m>U(8)</m> is cyclic.
</p>
</li>
<li>
<p>
<m>{\mathbb Q}</m> is cyclic.
</p>
</li>
<li>
<p>
If every proper subgroup of a group <m>G</m> is cyclic,
then <m>G</m> is a cyclic group.
</p>
</li>
<li>
<p>
A group with a finite number of subgroups is finite.
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="2">
<statement>
<p>
Find the order of each of the following elements.
</p>
<ol cols="2">
<li>
<p>
<m>5 \in {\mathbb Z}_{12}</m>
</p>
</li>
<li>
<p>
<m>\sqrt{3} \in {\mathbb R}</m>
</p>
</li>
<li>
<p>
<m>\sqrt{3} \in {\mathbb R}^\ast</m>
</p>
</li>
<li>
<p>
<m>-i \in {\mathbb C}^\ast</m>
</p>
</li>
<li>
<p>
72 in <m>{\mathbb Z}_{240}</m>
</p>
</li>
<li>
<p>
312 in <m>{\mathbb Z}_{471}</m>
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="3">
<statement>
<p>
List all of the elements in each of the following subgroups.
</p>
<ol>
<li>
<p>
The subgroup of <m>{\mathbb Z}</m> generated by 7
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb Z}_{24}</m> generated by 15
</p>
</li>
<li>
<p>
All subgroups of <m>{\mathbb Z}_{12}</m>
</p>
</li>
<li>
<p>
All subgroups of <m>{\mathbb Z}_{60}</m>
</p>
</li>
<li>
<p>
All subgroups of <m>{\mathbb Z}_{13}</m>
</p>
</li>
<li>
<p>
All subgroups of <m>{\mathbb Z}_{48}</m>
</p>
</li>
<li>
<p>
The subgroup generated by 3 in <m>U(20)</m>
</p>
</li>
<li>
<p>
The subgroup generated by 5 in <m>U(18)</m>
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb R}^\ast</m> generated by 7
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb C}^\ast</m> generated by <m>i</m> where <m>i^2 = -1</m>
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb C}^\ast</m> generated by <m>2i</m>
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb C}^\ast</m> generated by <m>(1 + i) / \sqrt{2}</m>
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb C}^\ast</m> generated by <m>(1 + \sqrt{3}\, i) / 2</m>
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="4">
<statement>
<p>
Find the subgroups of <m>GL_2( {\mathbb R })</m> generated by each of the following matrices.
</p>
<ol cols="3">
<li>
<p>
<m>\displaystyle \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}</m>
</p>
</li>
<li>
<p>
<m>\displaystyle \begin{pmatrix} 0 & 1/3 \\ 3 & 0 \end{pmatrix}</m>
</p>
</li>
<li>
<p>
<m>\displaystyle \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}</m>
</p>
</li>
<li>
<p>
<m>\displaystyle \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}</m>
</p>
</li>
<li>
<p>
<m>\displaystyle \begin{pmatrix} 1 & -1 \\ -1 & 0 \end{pmatrix}</m>
</p>
</li>
<li>
<p>
<m>\displaystyle \begin{pmatrix} \sqrt{3}/ 2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}</m>
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="5">
<statement>
<p>
Find the order of every element in <m>{\mathbb Z}_{18}</m>.
</p>
</statement>
</exercise>
<exercise number="6">
<statement>
<p>
Find the order of every element in the symmetry group of the square,
<m>D_4</m>.
</p>
</statement>
</exercise>
<exercise number="7">
<statement>
<p>
What are all of the cyclic subgroups of the quaternion group,
<m>Q_8</m>?
</p>
</statement>
</exercise>
<exercise number="8">
<statement>
<p>
List all of the cyclic subgroups of <m>U(30)</m>.
</p>
</statement>
</exercise>
<exercise number="9">
<statement>
<p>
List every generator of each subgroup of order 8 in <m>{\mathbb Z}_{32}</m>.
</p>
</statement>
</exercise>
<exercise number="10">
<statement>
<p>
Find all elements of finite order in each of the following groups.
Here the
<q><m>\ast</m></q>
indicates the set with zero removed.
</p>
<ol cols="3">
<li>
<p>
<m>{\mathbb Z}</m>
</p>
</li>
<li>
<p>
<m>{\mathbb Q}^\ast</m>
</p>
</li>
<li>
<p>
<m>{\mathbb R}^\ast</m>
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="11">
<statement>
<p>
If <m>a^{24} =e</m> in a group <m>G</m>,
what are the possible orders of <m>a</m>?
</p>
</statement>
</exercise>
<exercise number="12">
<statement>
<p>
Find a cyclic group with exactly one generator.
Can you find cyclic groups with exactly two generators?
Four generators?
How about <m>n</m> generators?
</p>
</statement>
</exercise>
<exercise number="13">
<statement>
<p>
For <m>n \leq 20</m>, which groups <m>U(n)</m> are cyclic?
Make a conjecture as to what is true in general.
Can you prove your conjecture?
</p>
</statement>
</exercise>
<exercise number="14">
<statement>
<p>
Let
<me>
A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \qquad \text{and} \qquad B = \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}
</me>
be elements in <m>GL_2( {\mathbb R} )</m>.
Show that <m>A</m> and <m>B</m> have finite orders but <m>AB</m> does not.
</p>
</statement>
</exercise>
<exercise number="15">
<statement>
<p>
Evaluate each of the following.
</p>
<ol cols="2">
<li>
<p>
<m>(3-2i)+ (5i-6)</m>
</p>
</li>
<li>
<p>
<m>(4-5i)-\overline{(4i -4)}</m>
</p>
</li>
<li>
<p>
<m>(5-4i)(7+2i)</m>
</p>
</li>
<li>
<p>
<m>(9-i) \overline{(9-i)}</m>
</p>
</li>
<li>
<p>
<m>i^{45}</m>
</p>
</li>
<li>
<p>
<m>(1+i)+\overline{(1+i)}</m>
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="16">
<statement>
<p>
Convert the following complex numbers to the form <m>a + bi</m>.
</p>
<ol cols="2">
<li>
<p>
<m>2 \cis(\pi / 6 )</m>
</p>
</li>
<li>
<p>
<m>5 \cis(9\pi/4)</m>
</p>
</li>
<li>
<p>
<m>3 \cis(\pi)</m>
</p>
</li>
<li>
<p>
<m>\cis(7\pi/4) /2</m>
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="17">
<statement>
<p>
Change the following complex numbers to polar representation.
</p>
<ol cols="3">
<li>
<p>
<m>1-i</m>
</p>
</li>
<li>
<p>
<m>-5</m>
</p>
</li>
<li>
<p>
<m>2+2i</m>
</p>
</li>
<li>
<p>
<m>\sqrt{3} + i</m>
</p>
</li>
<li>
<p>
<m>-3i</m>
</p>
</li>
<li>
<p>
<m>2i + 2 \sqrt{3}</m>
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="18">
<statement>
<p>
Calculate each of the following expressions.
</p>
<ol cols="2">
<li>
<p>
<m>(1+i)^{-1}</m>
</p>
</li>
<li>
<p>
<m>(1 - i)^{6}</m>
</p>
</li>
<li>
<p>
<m>(\sqrt{3} + i)^{5}</m>
</p>
</li>
<li>
<p>
<m>(-i)^{10}</m>
</p>
</li>
<li>
<p>
<m>((1-i)/2)^{4}</m>
</p>
</li>
<li>
<p>
<m>(-\sqrt{2} - \sqrt{2}\, i)^{12}</m>
</p>
</li>
<li>
<p>
<m>(-2 + 2i)^{-5}</m>
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="19">
<statement>
<p>
Prove each of the following statements.
</p>
<ol cols="2">
<li>
<p>
<m>|z| = | \overline{z}|</m>
</p>
</li>
<li>
<p>
<m>z \overline{z} = |z|^2</m>
</p>
</li>
<li>
<p>
<m>z^{-1} = \overline{z} / |z|^2</m>
</p>
</li>
<li>
<p>
<m>|z +w| \leq |z| + |w|</m>
</p>
</li>
<li>
<p>
<m>|z - w| \geq | |z| - |w||</m>
</p>
</li>
<li>
<p>
<m>|z w| = |z| |w|</m>
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="20">
<statement>
<p>
List and graph the 6th roots of unity.
What are the generators of this group?
What are the primitive 6th roots of unity?
</p>
</statement>
</exercise>
<exercise number="21">
<statement>
<p>
List and graph the 5th roots of unity.
What are the generators of this group?
What are the primitive 5th roots of unity?
</p>
</statement>
</exercise>
<exercise number="22">
<statement>
<p>
Calculate each of the following.
</p>
<ol cols="2">
<li>
<p>
<m>292^{3171} \pmod{ 582}</m>
</p>
</li>
<li>
<p>
<m>2557^{ 341} \pmod{ 5681}</m>
</p>
</li>
<li>
<p>
<m>2071^{ 9521} \pmod{ 4724}</m>
</p>
</li>
<li>
<p>
<m>971^{ 321} \pmod{ 765}</m>
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="23">
<statement>
<p>
Let <m>a, b \in G</m>.
Prove the following statements.
</p>
<ol>
<li>
<p>
The order of <m>a</m> is the same as the order of <m>a^{-1}</m>.
</p>
</li>
<li>
<p>
For all <m>g \in G</m>, <m>|a| = |g^{-1}ag|</m>.
</p>
</li>
<li>
<p>
The order of <m>ab</m> is the same as the order of <m>ba</m>.
</p>
</li>
</ol>
</statement>
</exercise>
<exercise number="24">
<statement>
<p>
Let <m>p</m> and <m>q</m> be distinct primes.
How many generators does <m>{\mathbb Z}_{pq}</m> have?
</p>
</statement>
</exercise>
<exercise number="25">
<statement>
<p>
Let <m>p</m> be prime and <m>r</m> be a positive integer.
How many generators does <m>{\mathbb Z}_{p^r}</m> have?
</p>
</statement>
</exercise>
<exercise number="26">
<statement>
<p>
Prove that <m>{\mathbb Z}_{p}</m> has no nontrivial subgroups if <m>p</m> is prime.
</p>
</statement>
</exercise>
<exercise number="27">
<statement>
<p>
If <m>g</m> and <m>h</m> have orders 15 and 16 respectively in a group <m>G</m>,
what is the order of <m>\langle g \rangle \cap \langle h \rangle </m>?
</p>
</statement>
</exercise>
<exercise number="28">
<statement>
<p>
Let <m>a</m> be an element in a group <m>G</m>.
What is a generator for the subgroup <m>\langle a^m \rangle \cap \langle a^n \rangle</m>?
</p>
</statement>
</exercise>
<exercise number="29">
<statement>
<p>
Prove that <m>{\mathbb Z}_n</m> has an even number of generators for <m>n \gt 2</m>.
</p>
</statement>
</exercise>
<exercise number="30">
<statement>
<p>
Suppose that <m>G</m> is a group and let <m>a</m>, <m>b \in G</m>.
Prove that if <m>|a| = m</m> and <m>|b| = n</m> with <m>\gcd(m,n) = 1</m>,
then <m>\langle a \rangle \cap \langle b \rangle = \{ e \}</m>.
</p>
</statement>
</exercise>
<exercise number="31">
<statement>
<p>
Let <m>G</m> be an abelian group.
Show that the elements of finite order in <m>G</m> form a subgroup.
This subgroup is called the <term>torsion subgroup</term> of <m>G</m>.
</p>
</statement>
</exercise>
<exercise number="32">
<statement>
<p>
Let <m>G</m> be a finite cyclic group of order <m>n</m> generated by <m>x</m>.
Show that if <m>y = x^k</m> where <m>\gcd(k,n) = 1</m>,
then <m>y</m> must be a generator of <m>G</m>.
</p>
</statement>
</exercise>
<exercise number="33">
<statement>
<p>
If <m>G</m> is an abelian group that contains a pair of cyclic subgroups of order 2, show that <m>G</m> must contain a subgroup of order 4.
Does this subgroup have to be cyclic?
</p>
</statement>
</exercise>
<exercise number="34">
<statement>
<p>
Let <m>G</m> be an abelian group of order <m>pq</m> where <m>\gcd(p,q) = 1</m>.
If <m>G</m> contains elements <m>a</m> and <m>b</m> of order <m>p</m> and <m>q</m> respectively,
then show that <m>G</m> is cyclic.
</p>
</statement>
</exercise>
<exercise number="35">
<statement>
<p>
Prove that the subgroups of <m>\mathbb Z</m> are exactly
<m>n{\mathbb Z}</m> for <m>n = 0, 1, 2, \ldots</m>.
</p>
</statement>
</exercise>
<exercise number="36">
<statement>
<p>
Prove that the generators of
<m>{\mathbb Z}_n</m> are the integers <m>r</m> such that
<m>1 \leq r \lt n</m> and <m>\gcd(r,n) = 1</m>.
</p>
</statement>
</exercise>
<exercise number="37">
<statement>
<p>
Prove that if <m>G</m> has no proper nontrivial subgroups,
then <m>G</m> is a cyclic group.
</p>
</statement>
</exercise>
<exercise number="38">
<statement>
<p>
Prove that the order of an element in a cyclic group <m>G</m> must divide the order of the group.
</p>
</statement>
</exercise>
<exercise number="39" xml:id="cyclic-exercise-subgroups-exist">
<statement>
<p>
Prove that if <m>G</m> is a cyclic group of order <m>m</m> and <m>d \mid m</m>,
then <m>G</m> must have a subgroup of order <m>d</m>.
</p>
</statement>
</exercise>
<exercise number="40">
<statement>
<p>
For what integers <m>n</m> is <m>-1</m> an <m>n</m>th root of unity?
</p>
</statement>
</exercise>
<exercise number="41">
<statement>
<p>
If <m>z = r( \cos \theta + i \sin \theta)</m> and
<m>w = s(\cos \phi + i \sin \phi)</m> are two nonzero complex numbers,
show that
<me>
zw = rs[ \cos( \theta + \phi) + i \sin( \theta + \phi)]
</me>.
</p>
</statement>
</exercise>
<exercise number="42">
<statement>
<p>
Prove that the circle group is a subgroup of <m>{\mathbb C}^*</m>.
</p>
</statement>
</exercise>
<exercise number="43">
<statement>
<p>
Prove that the <m>n</m>th roots of unity form a cyclic subgroup of <m>{\mathbb T}</m> of order <m>n</m>.
</p>
</statement>
</exercise>
<exercise number="44">
<statement>
<p>
Let <m>\alpha \in \mathbb T</m>.
Prove that <m>\alpha^m =1</m> and <m>\alpha^n = 1</m> if and only if
<m>\alpha^d = 1</m> for <m>d = \gcd(m,n)</m>.
</p>
</statement>
</exercise>
<exercise number="45">
<statement>
<p>
Let <m>z \in {\mathbb C}^\ast</m>.
If <m>|z| \neq 1</m>, prove that the order of <m>z</m> is infinite.
</p>
</statement>
</exercise>
<exercise number="46">
<statement>
<p>
Let <m>z =\cos \theta + i \sin \theta</m> be in
<m>{\mathbb T}</m> where <m>\theta \in {\mathbb Q}</m>.
Prove that the order of <m>z</m> is infinite.
</p>
</statement>
</exercise>
</exercises>
Exercises 2.5 Exercises
View Source for exercises
1.
View Source for exercise
<exercise number="1">
<statement>
<p>
Prove or disprove each of the following statements.
</p>
<ol>
<li>
<p>
All of the generators of <m>{\mathbb Z}_{60}</m> are prime.
</p>
</li>
<li>
<p>
<m>U(8)</m> is cyclic.
</p>
</li>
<li>
<p>
<m>{\mathbb Q}</m> is cyclic.
</p>
</li>
<li>
<p>
If every proper subgroup of a group <m>G</m> is cyclic,
then <m>G</m> is a cyclic group.
</p>
</li>
<li>
<p>
A group with a finite number of subgroups is finite.
</p>
</li>
</ol>
</statement>
</exercise>
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.
2.
View Source for exercise
<exercise number="2">
<statement>
<p>
Find the order of each of the following elements.
</p>
<ol cols="2">
<li>
<p>
<m>5 \in {\mathbb Z}_{12}</m>
</p>
</li>
<li>
<p>
<m>\sqrt{3} \in {\mathbb R}</m>
</p>
</li>
<li>
<p>
<m>\sqrt{3} \in {\mathbb R}^\ast</m>
</p>
</li>
<li>
<p>
<m>-i \in {\mathbb C}^\ast</m>
</p>
</li>
<li>
<p>
72 in <m>{\mathbb Z}_{240}</m>
</p>
</li>
<li>
<p>
312 in <m>{\mathbb Z}_{471}</m>
</p>
</li>
</ol>
</statement>
</exercise>
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 \({\mathbb Z}_{471}\)
3.
View Source for exercise
<exercise number="3">
<statement>
<p>
List all of the elements in each of the following subgroups.
</p>
<ol>
<li>
<p>
The subgroup of <m>{\mathbb Z}</m> generated by 7
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb Z}_{24}</m> generated by 15
</p>
</li>
<li>
<p>
All subgroups of <m>{\mathbb Z}_{12}</m>
</p>
</li>
<li>
<p>
All subgroups of <m>{\mathbb Z}_{60}</m>
</p>
</li>
<li>
<p>
All subgroups of <m>{\mathbb Z}_{13}</m>
</p>
</li>
<li>
<p>
All subgroups of <m>{\mathbb Z}_{48}</m>
</p>
</li>
<li>
<p>
The subgroup generated by 3 in <m>U(20)</m>
</p>
</li>
<li>
<p>
The subgroup generated by 5 in <m>U(18)</m>
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb R}^\ast</m> generated by 7
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb C}^\ast</m> generated by <m>i</m> where <m>i^2 = -1</m>
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb C}^\ast</m> generated by <m>2i</m>
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb C}^\ast</m> generated by <m>(1 + i) / \sqrt{2}</m>
</p>
</li>
<li>
<p>
The subgroup of <m>{\mathbb C}^\ast</m> generated by <m>(1 + \sqrt{3}\, i) / 2</m>
</p>
</li>
</ol>
</statement>
</exercise>
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 \((1 + \sqrt{3}\, i) / 2\)
4.
View Source for exercise
<exercise number="4">
<statement>
<p>
Find the subgroups of <m>GL_2( {\mathbb R })</m> generated by each of the following matrices.
</p>
<ol cols="3">
<li>
<p>
<m>\displaystyle \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}</m>
</p>
</li>
<li>
<p>
<m>\displaystyle \begin{pmatrix} 0 & 1/3 \\ 3 & 0 \end{pmatrix}</m>
</p>
</li>
<li>
<p>
<m>\displaystyle \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}</m>
</p>
</li>
<li>
<p>
<m>\displaystyle \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}</m>
</p>
</li>
<li>
<p>
<m>\displaystyle \begin{pmatrix} 1 & -1 \\ -1 & 0 \end{pmatrix}</m>
</p>
</li>
<li>
<p>
<m>\displaystyle \begin{pmatrix} \sqrt{3}/ 2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}</m>
</p>
</li>
</ol>
</statement>
</exercise>
Find the subgroups of \(GL_2( {\mathbb R })\) generated by each of the following matrices.
- \(\displaystyle \displaystyle \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\)
- \(\displaystyle \displaystyle \begin{pmatrix} 0 & 1/3 \\ 3 & 0 \end{pmatrix}\)
- \(\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}\)
- \(\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}\)
- \(\displaystyle \displaystyle \begin{pmatrix} 1 & -1 \\ -1 & 0 \end{pmatrix}\)
- \(\displaystyle \displaystyle \begin{pmatrix} \sqrt{3}/ 2 & 1/2 \\ -1/2 & \sqrt{3}/2 \end{pmatrix}\)
5.
View Source for exercise
<exercise number="5">
<statement>
<p>
Find the order of every element in <m>{\mathbb Z}_{18}</m>.
</p>
</statement>
</exercise>
Find the order of every element in \({\mathbb Z}_{18}\text{.}\)
6.
View Source for exercise
<exercise number="6">
<statement>
<p>
Find the order of every element in the symmetry group of the square,
<m>D_4</m>.
</p>
</statement>
</exercise>
Find the order of every element in the symmetry group of the square, \(D_4\text{.}\)
7.
View Source for exercise
<exercise number="7">
<statement>
<p>
What are all of the cyclic subgroups of the quaternion group,
<m>Q_8</m>?
</p>
</statement>
</exercise>
What are all of the cyclic subgroups of the quaternion group, \(Q_8\text{?}\)
8.
View Source for exercise
<exercise number="8">
<statement>
<p>
List all of the cyclic subgroups of <m>U(30)</m>.
</p>
</statement>
</exercise>
List all of the cyclic subgroups of \(U(30)\text{.}\)
9.
View Source for exercise
<exercise number="9">
<statement>
<p>
List every generator of each subgroup of order 8 in <m>{\mathbb Z}_{32}</m>.
</p>
</statement>
</exercise>
List every generator of each subgroup of order 8 in \({\mathbb Z}_{32}\text{.}\)
10.
View Source for exercise
<exercise number="10">
<statement>
<p>
Find all elements of finite order in each of the following groups.
Here the
<q><m>\ast</m></q>
indicates the set with zero removed.
</p>
<ol cols="3">
<li>
<p>
<m>{\mathbb Z}</m>
</p>
</li>
<li>
<p>
<m>{\mathbb Q}^\ast</m>
</p>
</li>
<li>
<p>
<m>{\mathbb R}^\ast</m>
</p>
</li>
</ol>
</statement>
</exercise>
Find all elements of finite order in each of the following groups. Here the “\(\ast\)” indicates the set with zero removed.
- \(\displaystyle {\mathbb Z}\)
- \(\displaystyle {\mathbb Q}^\ast\)
- \(\displaystyle {\mathbb R}^\ast\)
11.
View Source for exercise
<exercise number="11">
<statement>
<p>
If <m>a^{24} =e</m> in a group <m>G</m>,
what are the possible orders of <m>a</m>?
</p>
</statement>
</exercise>
If \(a^{24} =e\) in a group \(G\text{,}\) what are the possible orders of \(a\text{?}\)
12.
View Source for exercise
<exercise number="12">
<statement>
<p>
Find a cyclic group with exactly one generator.
Can you find cyclic groups with exactly two generators?
Four generators?
How about <m>n</m> generators?
</p>
</statement>
</exercise>
Find a cyclic group with exactly one generator. Can you find cyclic groups with exactly two generators? Four generators? How about \(n\) generators?
13.
View Source for exercise
<exercise number="13">
<statement>
<p>
For <m>n \leq 20</m>, which groups <m>U(n)</m> are cyclic?
Make a conjecture as to what is true in general.
Can you prove your conjecture?
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="14">
<statement>
<p>
Let
<me>
A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \qquad \text{and} \qquad B = \begin{pmatrix} 0 & -1 \\ 1 & -1 \end{pmatrix}
</me>
be elements in <m>GL_2( {\mathbb R} )</m>.
Show that <m>A</m> and <m>B</m> have finite orders but <m>AB</m> does not.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="15">
<statement>
<p>
Evaluate each of the following.
</p>
<ol cols="2">
<li>
<p>
<m>(3-2i)+ (5i-6)</m>
</p>
</li>
<li>
<p>
<m>(4-5i)-\overline{(4i -4)}</m>
</p>
</li>
<li>
<p>
<m>(5-4i)(7+2i)</m>
</p>
</li>
<li>
<p>
<m>(9-i) \overline{(9-i)}</m>
</p>
</li>
<li>
<p>
<m>i^{45}</m>
</p>
</li>
<li>
<p>
<m>(1+i)+\overline{(1+i)}</m>
</p>
</li>
</ol>
</statement>
</exercise>
Evaluate each of the following.
- \(\displaystyle (3-2i)+ (5i-6)\)
- \(\displaystyle (4-5i)-\overline{(4i -4)}\)
- \(\displaystyle (5-4i)(7+2i)\)
- \(\displaystyle (9-i) \overline{(9-i)}\)
- \(\displaystyle i^{45}\)
- \(\displaystyle (1+i)+\overline{(1+i)}\)
16.
View Source for exercise
<exercise number="16">
<statement>
<p>
Convert the following complex numbers to the form <m>a + bi</m>.
</p>
<ol cols="2">
<li>
<p>
<m>2 \cis(\pi / 6 )</m>
</p>
</li>
<li>
<p>
<m>5 \cis(9\pi/4)</m>
</p>
</li>
<li>
<p>
<m>3 \cis(\pi)</m>
</p>
</li>
<li>
<p>
<m>\cis(7\pi/4) /2</m>
</p>
</li>
</ol>
</statement>
</exercise>
Convert the following complex numbers to the form \(a + bi\text{.}\)
- \(\displaystyle 2 \cis(\pi / 6 )\)
- \(\displaystyle 5 \cis(9\pi/4)\)
- \(\displaystyle 3 \cis(\pi)\)
- \(\displaystyle \cis(7\pi/4) /2\)
17.
View Source for exercise
<exercise number="17">
<statement>
<p>
Change the following complex numbers to polar representation.
</p>
<ol cols="3">
<li>
<p>
<m>1-i</m>
</p>
</li>
<li>
<p>
<m>-5</m>
</p>
</li>
<li>
<p>
<m>2+2i</m>
</p>
</li>
<li>
<p>
<m>\sqrt{3} + i</m>
</p>
</li>
<li>
<p>
<m>-3i</m>
</p>
</li>
<li>
<p>
<m>2i + 2 \sqrt{3}</m>
</p>
</li>
</ol>
</statement>
</exercise>
Change the following complex numbers to polar representation.
- \(\displaystyle 1-i\)
- \(\displaystyle -5\)
- \(\displaystyle 2+2i\)
- \(\displaystyle \sqrt{3} + i\)
- \(\displaystyle -3i\)
- \(\displaystyle 2i + 2 \sqrt{3}\)
18.
View Source for exercise
<exercise number="18">
<statement>
<p>
Calculate each of the following expressions.
</p>
<ol cols="2">
<li>
<p>
<m>(1+i)^{-1}</m>
</p>
</li>
<li>
<p>
<m>(1 - i)^{6}</m>
</p>
</li>
<li>
<p>
<m>(\sqrt{3} + i)^{5}</m>
</p>
</li>
<li>
<p>
<m>(-i)^{10}</m>
</p>
</li>
<li>
<p>
<m>((1-i)/2)^{4}</m>
</p>
</li>
<li>
<p>
<m>(-\sqrt{2} - \sqrt{2}\, i)^{12}</m>
</p>
</li>
<li>
<p>
<m>(-2 + 2i)^{-5}</m>
</p>
</li>
</ol>
</statement>
</exercise>
Calculate each of the following expressions.
- \(\displaystyle (1+i)^{-1}\)
- \(\displaystyle (1 - i)^{6}\)
- \(\displaystyle (\sqrt{3} + i)^{5}\)
- \(\displaystyle (-i)^{10}\)
- \(\displaystyle ((1-i)/2)^{4}\)
- \(\displaystyle (-\sqrt{2} - \sqrt{2}\, i)^{12}\)
- \(\displaystyle (-2 + 2i)^{-5}\)
19.
View Source for exercise
<exercise number="19">
<statement>
<p>
Prove each of the following statements.
</p>
<ol cols="2">
<li>
<p>
<m>|z| = | \overline{z}|</m>
</p>
</li>
<li>
<p>
<m>z \overline{z} = |z|^2</m>
</p>
</li>
<li>
<p>
<m>z^{-1} = \overline{z} / |z|^2</m>
</p>
</li>
<li>
<p>
<m>|z +w| \leq |z| + |w|</m>
</p>
</li>
<li>
<p>
<m>|z - w| \geq | |z| - |w||</m>
</p>
</li>
<li>
<p>
<m>|z w| = |z| |w|</m>
</p>
</li>
</ol>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="20">
<statement>
<p>
List and graph the 6th roots of unity.
What are the generators of this group?
What are the primitive 6th roots of unity?
</p>
</statement>
</exercise>
List and graph the 6th roots of unity. What are the generators of this group? What are the primitive 6th roots of unity?
21.
View Source for exercise
<exercise number="21">
<statement>
<p>
List and graph the 5th roots of unity.
What are the generators of this group?
What are the primitive 5th roots of unity?
</p>
</statement>
</exercise>
List and graph the 5th roots of unity. What are the generators of this group? What are the primitive 5th roots of unity?
22.
View Source for exercise
<exercise number="22">
<statement>
<p>
Calculate each of the following.
</p>
<ol cols="2">
<li>
<p>
<m>292^{3171} \pmod{ 582}</m>
</p>
</li>
<li>
<p>
<m>2557^{ 341} \pmod{ 5681}</m>
</p>
</li>
<li>
<p>
<m>2071^{ 9521} \pmod{ 4724}</m>
</p>
</li>
<li>
<p>
<m>971^{ 321} \pmod{ 765}</m>
</p>
</li>
</ol>
</statement>
</exercise>
Calculate each of the following.
- \(\displaystyle 292^{3171} \pmod{ 582}\)
- \(\displaystyle 2557^{ 341} \pmod{ 5681}\)
- \(\displaystyle 2071^{ 9521} \pmod{ 4724}\)
- \(\displaystyle 971^{ 321} \pmod{ 765}\)
23.
View Source for exercise
<exercise number="23">
<statement>
<p>
Let <m>a, b \in G</m>.
Prove the following statements.
</p>
<ol>
<li>
<p>
The order of <m>a</m> is the same as the order of <m>a^{-1}</m>.
</p>
</li>
<li>
<p>
For all <m>g \in G</m>, <m>|a| = |g^{-1}ag|</m>.
</p>
</li>
<li>
<p>
The order of <m>ab</m> is the same as the order of <m>ba</m>.
</p>
</li>
</ol>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="24">
<statement>
<p>
Let <m>p</m> and <m>q</m> be distinct primes.
How many generators does <m>{\mathbb Z}_{pq}</m> have?
</p>
</statement>
</exercise>
Let \(p\) and \(q\) be distinct primes. How many generators does \({\mathbb Z}_{pq}\) have?
25.
View Source for exercise
<exercise number="25">
<statement>
<p>
Let <m>p</m> be prime and <m>r</m> be a positive integer.
How many generators does <m>{\mathbb Z}_{p^r}</m> have?
</p>
</statement>
</exercise>
Let \(p\) be prime and \(r\) be a positive integer. How many generators does \({\mathbb Z}_{p^r}\) have?
26.
View Source for exercise
<exercise number="26">
<statement>
<p>
Prove that <m>{\mathbb Z}_{p}</m> has no nontrivial subgroups if <m>p</m> is prime.
</p>
</statement>
</exercise>
Prove that \({\mathbb Z}_{p}\) has no nontrivial subgroups if \(p\) is prime.
27.
View Source for exercise
<exercise number="27">
<statement>
<p>
If <m>g</m> and <m>h</m> have orders 15 and 16 respectively in a group <m>G</m>,
what is the order of <m>\langle g \rangle \cap \langle h \rangle </m>?
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="28">
<statement>
<p>
Let <m>a</m> be an element in a group <m>G</m>.
What is a generator for the subgroup <m>\langle a^m \rangle \cap \langle a^n \rangle</m>?
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="29">
<statement>
<p>
Prove that <m>{\mathbb Z}_n</m> has an even number of generators for <m>n \gt 2</m>.
</p>
</statement>
</exercise>
Prove that \({\mathbb Z}_n\) has an even number of generators for \(n \gt 2\text{.}\)
30.
View Source for exercise
<exercise number="30">
<statement>
<p>
Suppose that <m>G</m> is a group and let <m>a</m>, <m>b \in G</m>.
Prove that if <m>|a| = m</m> and <m>|b| = n</m> with <m>\gcd(m,n) = 1</m>,
then <m>\langle a \rangle \cap \langle b \rangle = \{ e \}</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="31">
<statement>
<p>
Let <m>G</m> be an abelian group.
Show that the elements of finite order in <m>G</m> form a subgroup.
This subgroup is called the <term>torsion subgroup</term> of <m>G</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="32">
<statement>
<p>
Let <m>G</m> be a finite cyclic group of order <m>n</m> generated by <m>x</m>.
Show that if <m>y = x^k</m> where <m>\gcd(k,n) = 1</m>,
then <m>y</m> must be a generator of <m>G</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="33">
<statement>
<p>
If <m>G</m> is an abelian group that contains a pair of cyclic subgroups of order 2, show that <m>G</m> must contain a subgroup of order 4.
Does this subgroup have to be cyclic?
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="34">
<statement>
<p>
Let <m>G</m> be an abelian group of order <m>pq</m> where <m>\gcd(p,q) = 1</m>.
If <m>G</m> contains elements <m>a</m> and <m>b</m> of order <m>p</m> and <m>q</m> respectively,
then show that <m>G</m> is cyclic.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="35">
<statement>
<p>
Prove that the subgroups of <m>\mathbb Z</m> are exactly
<m>n{\mathbb Z}</m> for <m>n = 0, 1, 2, \ldots</m>.
</p>
</statement>
</exercise>
Prove that the subgroups of \(\mathbb Z\) are exactly \(n{\mathbb Z}\) for \(n = 0, 1, 2, \ldots\text{.}\)
36.
View Source for exercise
<exercise number="36">
<statement>
<p>
Prove that the generators of
<m>{\mathbb Z}_n</m> are the integers <m>r</m> such that
<m>1 \leq r \lt n</m> and <m>\gcd(r,n) = 1</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="37">
<statement>
<p>
Prove that if <m>G</m> has no proper nontrivial subgroups,
then <m>G</m> is a cyclic group.
</p>
</statement>
</exercise>
Prove that if \(G\) has no proper nontrivial subgroups, then \(G\) is a cyclic group.
38.
View Source for exercise
<exercise number="38">
<statement>
<p>
Prove that the order of an element in a cyclic group <m>G</m> must divide the order of the group.
</p>
</statement>
</exercise>
Prove that the order of an element in a cyclic group \(G\) must divide the order of the group.
39.
View Source for exercise
<exercise number="39" xml:id="cyclic-exercise-subgroups-exist">
<statement>
<p>
Prove that if <m>G</m> is a cyclic group of order <m>m</m> and <m>d \mid m</m>,
then <m>G</m> must have a subgroup of order <m>d</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="40">
<statement>
<p>
For what integers <m>n</m> is <m>-1</m> an <m>n</m>th root of unity?
</p>
</statement>
</exercise>
For what integers \(n\) is \(-1\) an \(n\)th root of unity?
41.
View Source for exercise
<exercise number="41">
<statement>
<p>
If <m>z = r( \cos \theta + i \sin \theta)</m> and
<m>w = s(\cos \phi + i \sin \phi)</m> are two nonzero complex numbers,
show that
<me>
zw = rs[ \cos( \theta + \phi) + i \sin( \theta + \phi)]
</me>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="42">
<statement>
<p>
Prove that the circle group is a subgroup of <m>{\mathbb C}^*</m>.
</p>
</statement>
</exercise>
Prove that the circle group is a subgroup of \({\mathbb C}^*\text{.}\)
43.
View Source for exercise
<exercise number="43">
<statement>
<p>
Prove that the <m>n</m>th roots of unity form a cyclic subgroup of <m>{\mathbb T}</m> of order <m>n</m>.
</p>
</statement>
</exercise>
Prove that the \(n\)th roots of unity form a cyclic subgroup of \({\mathbb T}\) of order \(n\text{.}\)
44.
View Source for exercise
<exercise number="44">
<statement>
<p>
Let <m>\alpha \in \mathbb T</m>.
Prove that <m>\alpha^m =1</m> and <m>\alpha^n = 1</m> if and only if
<m>\alpha^d = 1</m> for <m>d = \gcd(m,n)</m>.
</p>
</statement>
</exercise>
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.
View Source for exercise
<exercise number="45">
<statement>
<p>
Let <m>z \in {\mathbb C}^\ast</m>.
If <m>|z| \neq 1</m>, prove that the order of <m>z</m> is infinite.
</p>
</statement>
</exercise>
Let \(z \in {\mathbb C}^\ast\text{.}\) If \(|z| \neq 1\text{,}\) prove that the order of \(z\) is infinite.
46.
View Source for exercise
<exercise number="46">
<statement>
<p>
Let <m>z =\cos \theta + i \sin \theta</m> be in
<m>{\mathbb T}</m> where <m>\theta \in {\mathbb Q}</m>.
Prove that the order of <m>z</m> is infinite.
</p>
</statement>
</exercise>
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.
