PreTeXt Sample Book Abstract Algebra (SAMPLE ONLY)

<section>

  <title>Cyclic groups</title>

  <p>
    Often a subgroup will depend entirely on a single element of the group; that is, knowing that particular element will allow us to compute any other element in the subgroup.
  </p>

  <!-- RAB 2014/08/18 Maybe this id should have 3z, not z3? -->

  <example xml:id="example-cyclic-z3">

    <title>An Infinite Cyclic Subgroup, Modular Addition</title>

    <p>
      Suppose that we consider <m>3 \in {\mathbb Z}</m> and look at all multiples (both positive and negative) of 3.
      As a set, this is
      <md>
        3 {\mathbb Z} = \{ \ldots, -3, 0, 3, 6, \ldots \}
      </md>.

      It is easy to see that <m>3 {\mathbb Z}</m> is a subgroup of the integers.
      This subgroup is completely determined by the element 3 since we can obtain all of the other elements of the group by taking multiples of 3.
      Every element in the subgroup is <q>generated</q> by 3.
    </p>

  </example>

  <example xml:id="example-cyclic-2n">

    <title>An Infinite Cyclic Subgroup, Multiplication of Rational Numbers</title>

    <p>
      If <m>H = \{ 2^n : n \in {\mathbb Z} \}</m>, then <m>H</m> is a subgroup of the multiplicative group of nonzero rational numbers, <m>{\mathbb Q}^*</m>.
      If <m>a = 2^m</m> and <m>b = 2^n</m> are in <m>H</m>, then <m>ab^{-1} = 2^m 2^{-n} = 2^{m-n}</m> is also in <m>H</m>.
      By <xref ref="proposition-subgroup"/>, <m>H</m> is a subgroup of <m>{\mathbb Q}^*</m> determined by the element 2.
    </p>

  </example>

  <theorem xml:id="theorem-cyclic-subgroup">

    <statement>

      <p>
        Let <m>G</m> be a group and <m>a</m> be any element in <m>G</m>.
        Then the set
        <md>
          \langle a \rangle  = \{ a^k : k \in {\mathbb Z} \}
        </md>

        is a subgroup of <m>G</m>.
        Furthermore, <m>\langle a \rangle</m> is the smallest subgroup of <m>G</m> that contains~<m>a</m>.
        <notation>

          <usage><m>\langle a \rangle</m></usage>

          <description>
            cyclic group generated by <m>a</m>
          </description>

        </notation>

      </p>

    </statement>

    <proof>

      <p>
        The identity is in <m>\langle a \rangle </m> since <m>a^0 = e</m>.
        If <m>g</m> and <m>h</m> are any two elements in <m>\langle a \rangle </m>, then by the definition of <m>\langle a \rangle</m> we can write <m>g = a^m</m> and <m>h = a^n</m> for some integers <m>m</m> and <m>n</m>.
        So <m>gh = a^m a^n = a^{m+n}</m> is again in <m>\langle a \rangle </m>.
        Finally, if <m>g = a^n</m> in <m>\langle a \rangle </m>, then the inverse <m>g^{-1} = a^{-n}</m> is also in <m>\langle a \rangle </m>.
        Clearly, any subgroup <m>H</m> of <m>G</m> containing <m>a</m> must contain all the powers of <m>a</m> by closure; hence, <m>H</m> contains <m>\langle a \rangle </m>.
        Therefore, <m>\langle a \rangle </m> is the smallest subgroup of <m>G</m> containing <m>a</m>.
      </p>

    </proof>

  </theorem>

  <remark>

    <p>
      If we are using the <q>+</q> notation, as in the case of the integers under addition, we write <m>\langle a \rangle  = \{ na : n \in {\mathbb Z} \}</m>.
    </p>

  </remark>

  <p>
    For <m>a \in G</m>, we call <m>\langle a \rangle </m> the <term>cyclic subgroup</term><idx><h>Subgroup</h><h>cyclic</h></idx>
    generated by <m>a</m>.
    If <m>G</m> contains some element <m>a</m> such that <m>G = \langle a \rangle </m>, then <m>G</m> is a <term>cyclic group</term><idx><h>Group</h><h>cyclic</h></idx>.
    In this case <m>a</m> is a <term>generator</term><idx><h>Generator of a cyclic subgroup</h></idx>
    of <m>G</m>.
    If <m>a</m> is an element of a group <m>G</m>, we define the <term>order</term><idx><h>Element</h><h>order of</h></idx>
    of <m>a</m> to be the smallest positive integer <m>n</m> such that <m>a^n= e</m>, and we write <m>|a| = n</m>.
    <notation>

      <usage><m>|a|</m></usage>

      <description>
        the order of an element <m>a</m>
      </description>

    </notation>

    If there is no such integer <m>n</m>, we say that the order of <m>a</m> is infinite and  write <m>|a| = \infty</m> to denote the order of <m>a</m>.
  </p>

  <example xml:id="example-cyclic-z6">

    <title>Generators of a Finite Cyclic Group</title>

    <p>
      Notice that a cyclic group can have more than a single generator.
      Both 1 and 5 generate <m>{\mathbb Z}_6</m>; hence, <m>{\mathbb Z}_6</m> is a cyclic group.
      Not every element in a cyclic group is necessarily a generator of the group.
      The order of <m>2 \in {\mathbb Z}_6</m> is 3.
      The cyclic subgroup generated by 2 is <m>\langle 2 \rangle  = \{ 0, 2, 4 \}</m>.
    </p>

  </example>

  <p>
    The groups <m>{\mathbb Z}</m> and <m>{\mathbb Z}_n</m> are cyclic groups.
    The elements 1 and <m>-1</m> are generators for <m>{\mathbb Z}</m>.
    We can certainly generate <m>{\mathbb Z}_n</m> with 1 although there may be other generators of <m>{\mathbb Z}_n</m>, as in the case of <m>{\mathbb Z}_6</m>.
  </p>

  <example xml:id="example-cyclic-u9">

    <title>A Cyclic Group of Units</title>

    <p>
      The group of units, <m>U(9)</m>, in <m>{\mathbb Z}_9</m> is a cyclic group.
      As a set, <m>U(9)</m> is <m>\{ 1, 2, 4, 5, 7, 8  \}</m>.
      The element 2 is a generator for <m>U(9)</m> since
      <md>

        <mrow>2^1 &amp; = 2 \qquad 2^2 = 4</mrow>

        <mrow>2^3 &amp; = 8 \qquad 2^4 = 7</mrow>

        <mrow>2^5 &amp; =  5 \qquad 2^6 = 1</mrow>

      </md>.

    </p>

  </example>

  <example xml:id="example-cyclic-s3-not-cyclic">

    <title>A Group That is Not Cyclic</title>

    <p>
      Not every group is a cyclic group.
      Consider the symmetry group of an equilateral triangle <m>S_3</m>.
      The subgroups of <m>S_3</m> are shown in <xref ref="figure-subgrps-s3"/>.
      Notice that every subgroup is cyclic; however, no single element generates the entire group.
    </p>

  </example>

  <figure xml:id="figure-subgrps-s3">

    <caption>Subgroups of <m>S_3</m></caption>

    <!-- Replaced figure with tikz figure - TWJ 5/6/2010 -->

    <image xml:id="cyclic-s3-subgroups">

      <latex-image> \begin{tikzpicture}[scale=1] \draw  (0,0.3) -- (2.6,1.2); \draw  (2,0.3) -- (2.8,1.2); \draw  (4,0.3) -- (3.2,1.2); \draw  (6,0.3) -- (3.4,1.2); \draw  (0,-0.3) -- (2.6,-1.2); \draw  (2,-0.3) -- (2.8,-1.2); \draw  (4,-0.3) -- (3.2,-1.2); \draw  (6,-0.3) -- (3.4,-1.2); \node at (0, 0) {$\{  \identity, \rho_1, \rho_2\}$}; \node at (2, 0) {$\{  \identity, \mu_1\}$}; \node at (4, 0) {$\{  \identity, \mu_2 \}$}; \node at (6, 0) {$\{  \identity, \mu_3 \}$}; \node at (3, 1.5) {$S_3$}; \node at (3,-1.5) {$\{ \identity \}$}; \end{tikzpicture}

      </latex-image>

    </image>

  </figure>

  <theorem xml:id="theorem-cyclic-abelian">

    <statement>

      <p>
        Every cyclic group is abelian.
      </p>

    </statement>

    <proof>

      <p>
        Let <m>G</m> be a cyclic group and <m>a \in G</m> be a generator for <m>G</m>.
        If <m>g</m> and <m>h</m> are in <m>G</m>, then they can be written as powers of <m>a</m>, say <m>g = a^r</m> and <m>h = a^s</m>.
        Since
        <md>
          g  h = a^r a^s = a^{r+s} = a^{s+r} = a^s a^r = h g
        </md>,

        <m>G</m> is abelian.

      </p>

    </proof>

  </theorem>

</section>