<exercises xml:id="exercises-cyclic">
<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
<md>A =
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}
\qquad \text{and} \qquad
B =
\begin{pmatrix}
0 & -1 \\
1 & -1
\end{pmatrix}</md>
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>
<!-- TODO: Fix references to torsion subgroup -->
<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
<md>zw = rs[ \cos( \theta + \phi) + i \sin( \theta + \phi)].</md></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 4.5 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
<md>A =
\begin{pmatrix}
0 & 1 \\
-1 & 0
\end{pmatrix}
\qquad \text{and} \qquad
B =
\begin{pmatrix}
0 & -1 \\
1 & -1
\end{pmatrix}</md>
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
<md>zw = rs[ \cos( \theta + \phi) + i \sin( \theta + \phi)].</md></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)].
\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.
You have attempted of activities on this page.
