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PreTeXt Sample Book Abstract Algebra (SAMPLE ONLY)

Section 2.1 Cyclic groups

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  <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
        <me>
          3 {\mathbb Z} = \{ \ldots, -3, 0, 3, 6, \ldots \}
        </me>.
        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
          <me>
            \langle a \rangle  = \{ a^k : k \in {\mathbb Z} \}
          </me>
          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
          <me>
            g  h = a^r a^s = a^{r+s} = a^{s+r} = a^s a^r = h g
          </me>,
          <m>G</m> is abelian.
        </p>
      </proof>
    </theorem>
  </section>
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.

Example 2.1.1. An Infinite Cyclic Subgroup, Modular Addition.

View Source for example
<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
    <me>
      3 {\mathbb Z} = \{ \ldots, -3, 0, 3, 6, \ldots \}
    </me>.
    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>
Suppose that we consider \(3 \in {\mathbb Z}\) and look at all multiples (both positive and negative) of 3. As a set, this is
\begin{equation*} 3 {\mathbb Z} = \{ \ldots, -3, 0, 3, 6, \ldots \}\text{.} \end{equation*}
It is easy to see that \(3 {\mathbb Z}\) 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 β€œgenerated” by 3.

Example 2.1.2. An Infinite Cyclic Subgroup, Multiplication of Rational Numbers.

View Source for 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>
If \(H = \{ 2^n : n \in {\mathbb Z} \}\text{,}\) then \(H\) is a subgroup of the multiplicative group of nonzero rational numbers, \({\mathbb Q}^*\text{.}\) If \(a = 2^m\) and \(b = 2^n\) are in \(H\text{,}\) then \(ab^{-1} = 2^m 2^{-n} = 2^{m-n}\) is also in \(H\text{.}\) By PropositionΒ 1.3.8, \(H\) is a subgroup of \({\mathbb Q}^*\) determined by the element 2.

Proof.

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

Remark 2.1.4.

View Source for remark
<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>
If we are using the β€œ+” notation, as in the case of the integers under addition, we write \(\langle a \rangle = \{ na : n \in {\mathbb Z} \}\text{.}\)
For \(a \in G\text{,}\) we call \(\langle a \rangle \) the cyclic subgroup generated by \(a\text{.}\) If \(G\) contains some element \(a\) such that \(G = \langle a \rangle \text{,}\) then \(G\) is a cyclic group. In this case \(a\) is a generator of \(G\text{.}\) If \(a\) is an element of a group \(G\text{,}\) we define the order of \(a\) to be the smallest positive integer \(n\) such that \(a^n= e\text{,}\) and we write \(|a| = n\text{.}\) If there is no such integer \(n\text{,}\) we say that the order of \(a\) is infinite and write \(|a| = \infty\) to denote the order of \(a\text{.}\)

Example 2.1.5. Generators of a Finite Cyclic Group.

View Source for example
<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>
Notice that a cyclic group can have more than a single generator. Both 1 and 5 generate \({\mathbb Z}_6\text{;}\) hence, \({\mathbb Z}_6\) is a cyclic group. Not every element in a cyclic group is necessarily a generator of the group. The order of \(2 \in {\mathbb Z}_6\) is 3. The cyclic subgroup generated by 2 is \(\langle 2 \rangle = \{ 0, 2, 4 \}\text{.}\)
The groups \({\mathbb Z}\) and \({\mathbb Z}_n\) are cyclic groups. The elements 1 and \(-1\) are generators for \({\mathbb Z}\text{.}\) We can certainly generate \({\mathbb Z}_n\) with 1 although there may be other generators of \({\mathbb Z}_n\text{,}\) as in the case of \({\mathbb Z}_6\text{.}\)

Example 2.1.6. A Cyclic Group of Units.

View Source for example
<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>
The group of units, \(U(9)\text{,}\) in \({\mathbb Z}_9\) is a cyclic group. As a set, \(U(9)\) is \(\{ 1, 2, 4, 5, 7, 8 \}\text{.}\) The element 2 is a generator for \(U(9)\) since
\begin{align*} 2^1 & = 2 \qquad 2^2 = 4\\ 2^3 & = 8 \qquad 2^4 = 7\\ 2^5 & = 5 \qquad 2^6 = 1\text{.} \end{align*}

Example 2.1.7. A Group That is Not Cyclic.

View Source for 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>
Not every group is a cyclic group. Consider the symmetry group of an equilateral triangle \(S_3\text{.}\) The subgroups of \(S_3\) are shown in FigureΒ 2.1.8. Notice that every subgroup is cyclic; however, no single element generates the entire group.
View Source for figure
    <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>
Figure 2.1.8. Subgroups of \(S_3\)

Proof.

View Source for proof
<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
    <me>
      g  h = a^r a^s = a^{r+s} = a^{s+r} = a^s a^r = h g
    </me>,
    <m>G</m> is abelian.
  </p>
</proof>
Let \(G\) be a cyclic group and \(a \in G\) be a generator for \(G\text{.}\) If \(g\) and \(h\) are in \(G\text{,}\) then they can be written as powers of \(a\text{,}\) say \(g = a^r\) and \(h = a^s\text{.}\) Since
\begin{equation*} g h = a^r a^s = a^{r+s} = a^{s+r} = a^s a^r = h g\text{,} \end{equation*}
\(G\) is abelian.