PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

<exercises xml:id="groups-sage-exercises" xml:base="sage/groups-sage-exercises.xml">
  <title>Sage Exercises</title>
  <introduction>
    <p>
      These exercises are about becoming comfortable working with groups in Sage.
    </p>
  </introduction>
  <exercise number="1">
    <statement>
      <p>
        Create the groups <c>CyclicPermutationGroup(8)</c> and <c>DihedralGroup(4)</c> and name these groups <c>C</c> and <c>D</c>,
        respectively.
        We will understand these constructions better shortly,
        but for now just understand that both objects you create are actually groups.
      </p>
    </statement>
  </exercise>
  <exercise number="2">
    <statement>
      <p>
        Check that <c>C</c> and <c>D</c> have the same size by using the <c>.order()</c> method.
        Determine which group is abelian, and which is not,
        by using the <c>.is_abelian()</c> method.
      </p>
    </statement>
  </exercise>
  <exercise number="3">
    <statement>
      <p>
        Use the <c>.cayley_table()</c> method to create the Cayley table for each group.
      </p>
    </statement>
  </exercise>
  <exercise number="4">
    <statement>
      <p>
        Write a nicely formatted discussion identifying differences between the two groups that are discernible in properties of their Cayley tables.
        In other words,
        what is {\em different} about these two groups that you can
        <q>see</q>
        in the Cayley tables? (In the Sage notebook,
        a Shift-click on a blue bar will bring up a mini-word-processor,
        and you can use use dollar signs to embed mathematics formatted using <tex /> syntax.)
      </p>
    </statement>
  </exercise>
  <exercise number="5">
    <statement>
      <p>
        For <c>C</c> locate the one subgroup of order <m>4</m>.
        The group <c>D</c> has three subgroups of order <m>4</m>.
        Select one of the three subgroups of <c>D</c> that has a different structure than the subgroup you obtained from <c>C</c>.
      </p>
      <p>
        The <c>.subgroups()</c> method will give you a list of all of the subgroups to help you get started.
        A Cayley table will help you tell the difference between the two subgroups.
        What properties of these tables did you use to determine the difference in the structure of the subgroups?
      </p>
    </statement>
  </exercise>
  <exercise number="6">
    <statement>
      <p>
        The <c>.subgroup(elt_list)</c> method of a group will create the smallest subgroup containing the specified elements of the group,
        when given the elements as a list <c>elt_list</c>.
        Use this command to discover the shortest list of elements necessary to recreate the subgroups you found in the previous exercise.
        The equality comparison,
        <c>==</c>,
        can be used to test if two subgroups are equal.
      </p>
    </statement>
  </exercise>
</exercises>