Exercises 2.5 Exercises

  <exercises xml:id="exercises-cyclic" xmlns:pi="http://pretextbook.org/2020/pretext/internal">

<title>Exercises</title>

<exercisegroup>

<introduction>

  <p>
    Prove or disprove each of the following statements.
  </p>

</introduction>

<exercise>

  <statement>

    <p>
      All of the generators of <m>{\mathbb Z}_{60}</m> are prime.
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>U(8)</m> is cyclic.

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>{\mathbb Q}</m> is cyclic.

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      If every proper subgroup of a group <m>G</m> is cyclic, then <m>G</m> is a cyclic group.
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      A group with a finite number of subgroups is finite.
    </p>

  </statement>

</exercise>

</exercisegroup> <exercisegroup cols="2">

<introduction>

  <p>
    Find the order of each of the following elements.
  </p>

</introduction>

<exercise>

  <statement>

    <p>

      <m>5 \in {\mathbb Z}_{12}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>\sqrt{3} \in {\mathbb R}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>\sqrt{3} \in {\mathbb R}^\ast</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>-i \in {\mathbb C}^\ast</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      72 in <m>{\mathbb Z}_{240}</m>
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      312 in <m>{\mathbb Z}_{471}</m>
    </p>

  </statement>

</exercise>

</exercisegroup> <exercisegroup>

<introduction>

  <p>
    List all of the elements in each of the following subgroups.
  </p>

</introduction>

<exercise>

  <statement>

    <p>
      The subgroup of <m>{\mathbb Z}</m> generated by 7
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      The subgroup of <m>{\mathbb Z}_{24}</m> generated by 15
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      All subgroups of <m>{\mathbb Z}_{12}</m>
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      All subgroups of <m>{\mathbb Z}_{60}</m>
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      All subgroups of <m>{\mathbb Z}_{13}</m>
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      All subgroups of <m>{\mathbb Z}_{48}</m>
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      The subgroup generated by 3 in <m>U(20)</m>
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      The subgroup generated by 5 in <m>U(18)</m>
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      The subgroup of <m>{\mathbb R}^\ast</m> generated by 7
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      The subgroup of <m>{\mathbb C}^\ast</m> generated by <m>i</m> where <m>i^2 = -1</m>
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      The subgroup of <m>{\mathbb C}^\ast</m> generated by <m>2i</m>
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

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

  </statement>

</exercise>

<exercise>

  <statement>

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

  </statement>

</exercise>

</exercisegroup> <exercisegroup cols="3">

<introduction>

  <p>
    Find the subgroups of <m>GL_2( {\mathbb R })</m> generated by each of the following matrices.
  </p>

</introduction>

<exercise>

  <statement>

    <p>

      <m>\displaystyle \begin{pmatrix} 0 &amp; 1 \\ -1 &amp; 0 \end{pmatrix}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>\displaystyle \begin{pmatrix} 0 &amp; 1/3 \\ 3 &amp; 0 \end{pmatrix}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>\displaystyle \begin{pmatrix} 1 &amp; -1 \\ 1 &amp; 0 \end{pmatrix}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>\displaystyle \begin{pmatrix} 1 &amp; -1 \\ 0 &amp; 1 \end{pmatrix}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>\displaystyle \begin{pmatrix} 1 &amp; -1 \\ -1 &amp; 0 \end{pmatrix}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>\displaystyle \begin{pmatrix} \sqrt{3}/ 2 &amp; 1/2 \\ -1/2 &amp; \sqrt{3}/2 \end{pmatrix}</m>

    </p>

  </statement>

</exercise>

</exercisegroup>

<exercise>

  <statement>

    <p>
      Find the order of every element in <m>{\mathbb Z}_{18}</m>.
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      Find the order of every element in the symmetry group of the square, <m>D_4</m>.
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      What are all of the cyclic subgroups of the quaternion group, <m>Q_8</m>?
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      List all of the cyclic subgroups of <m>U(30)</m>.
    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>
      List every generator of each subgroup of order 8 in <m>{\mathbb Z}_{32}</m>.
    </p>

  </statement>

</exercise>

<exercisegroup cols="3">

<introduction>

  <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>

</introduction>

<exercise>

  <statement>

    <p>

      <m>{\mathbb Z}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>{\mathbb Q}^\ast</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>{\mathbb R}^\ast</m>

    </p>

  </statement>

</exercise>

</exercisegroup>

<exercise>

  <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>

  <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>

  <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>

  <statement>

    <p>
      Let
      <md>
        A = \begin{pmatrix} 0 &amp; 1 \\ -1 &amp; 0 \end{pmatrix} \qquad \text{and} \qquad B = \begin{pmatrix} 0 &amp; -1 \\ 1 &amp; -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>

<exercisegroup cols="2">

<introduction>

  <p>
    Evaluate each of the following.
  </p>

</introduction>

<exercise>

  <statement>

    <p>

      <m>(3-2i)+ (5i-6)</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>(4-5i)-\overline{(4i -4)}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>(5-4i)(7+2i)</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>(9-i) \overline{(9-i)}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>i^{45}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>(1+i)+\overline{(1+i)}</m>

    </p>

  </statement>

</exercise>

</exercisegroup> <exercisegroup cols="2">

<introduction>

  <p>
    Convert the following complex numbers to the form <m>a + bi</m>.
  </p>

</introduction>

<exercise>

  <statement>

    <p>

      <m>2 \cis(\pi / 6 )</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>5 \cis(9\pi/4)</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>3 \cis(\pi)</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>\cis(7\pi/4) /2</m>

    </p>

  </statement>

</exercise>

</exercisegroup> <exercisegroup cols="3">

<introduction>

  <p>
    Change the following complex numbers to polar representation.
  </p>

</introduction>

<exercise>

  <statement>

    <p>

      <m>1-i</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>-5</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>2+2i</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>\sqrt{3} + i</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>-3i</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>2i + 2 \sqrt{3}</m>

    </p>

  </statement>

</exercise>

</exercisegroup> <exercisegroup cols="2">

<introduction>

  <p>
    Calculate each of the following expressions.
  </p>

</introduction>

<exercise>

  <statement>

    <p>

      <m>(1+i)^{-1}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>(1 - i)^{6}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>(\sqrt{3} + i)^{5}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>(-i)^{10}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>((1-i)/2)^{4}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>(-\sqrt{2} - \sqrt{2}\, i)^{12}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>(-2 + 2i)^{-5}</m>

    </p>

  </statement>

</exercise>

</exercisegroup> <exercisegroup cols="2">

<introduction>

  <p>
    Prove each of the following statements.
  </p>

</introduction>

<exercise>

  <statement>

    <p>

      <m>|z| = | \overline{z}|</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>z \overline{z} = |z|^2</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>z^{-1} = \overline{z} / |z|^2</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>|z +w| \leq |z| + |w|</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>|z - w| \geq | |z| - |w||</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>|z w| = |z| |w|</m>

    </p>

  </statement>

</exercise>

</exercisegroup>

<exercise>

  <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>

  <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>

<exercisegroup cols="2">

<introduction>

  <p>
    Calculate each of the following.
  </p>

</introduction>

<exercise>

  <statement>

    <p>

      <m>292^{3171} \pmod{ 582}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>2557^{ 341} \pmod{ 5681}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>2071^{ 9521} \pmod{ 4724}</m>

    </p>

  </statement>

</exercise>

<exercise>

  <statement>

    <p>

      <m>971^{ 321} \pmod{ 765}</m>

    </p>

  </statement>

</exercise>

</exercisegroup>

<exercise>

  <introduction>

    <p>
      Let <m>a, b \in G</m>.
      Prove the following statements.
    </p>

  </introduction>

  <task>

    <statement>

      <p>
        The order of <m>a</m> is the same as the order of <m>a^{-1}</m>.
      </p>

    </statement>

  </task>

  <task>

    <statement>

      <p>
        For all <m>g \in G</m>, <m>|a| = |g^{-1}ag|</m>.
      </p>

    </statement>

  </task>

  <task>

    <statement>

      <p>
        The order of <m>ab</m> is the same as the order of <m>ba</m>.
      </p>

    </statement>

  </task>

</exercise>

<exercise>

  <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>

  <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>

  <statement>

    <p>
      Prove that  <m>{\mathbb Z}_{p}</m> has no nontrivial subgroups if <m>p</m> is prime.
    </p>

  </statement>

</exercise>

<exercise>

  <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>

  <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>

  <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>

  <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>

  <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>

  <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>

  <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>

  <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>

  <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>

  <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>

  <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>

  <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 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>

  <statement>

    <p>
      For what integers <m>n</m> is <m>-1</m> an <m>n</m>th root of unity?
    </p>

  </statement>

</exercise>

<exercise>

  <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>

  <statement>

    <p>
      Prove that the circle group is a subgroup of  <m>{\mathbb C}^*</m>.
    </p>

  </statement>

</exercise>

<exercise>

  <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>

  <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>

  <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>

  <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>