PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

<section xml:id="section-groups-define">
  <title>Definitions and Examples</title>


  <subsection>
    <title>Definition of a Group</title>
    <p>
      The integers mod <m>n</m> and the symmetries of a triangle or a rectangle are examples of groups.
      A <term>binary operation</term>
          <idx><h>Binary operation</h></idx>
      or <term>law of composition</term>
      on a set <m>G</m> is a function
      <m>G \times G \rightarrow G</m> that assigns to each pair
      <m>(a,b) \in G \times G</m> a unique element <m>a \circ b</m>,
      or <m>ab</m> in <m>G</m>, called the composition of <m>a</m> and <m>b</m>.
      A <term>group</term><idx><h>Group</h><h>definition of</h></idx>
      <m>(G, \circ )</m> is a set <m>G</m> together with a law of composition
      <m>(a,b) \mapsto a \circ b</m> that satisfies the following axioms.
      <ul>
        <li>
          <p>
            The law of composition is <term>associative</term>.
            That is,
            <me>
              (a \circ b) \circ c = a \circ (b \circ c)
            </me>
            for <m>a, b, c \in G</m>.
          </p>
        </li>
        <li>
          <p>
            There exists an element <m>e \in G</m>,
            called the <term>identity element</term>,
              <idx><h>Element</h><h>identity</h></idx>
            such that for any element <m>a \in G</m>
            <me>
              e \circ a = a \circ e = a
            </me>.
          </p>
        </li>
        <li>
          <p>
            For each element <m>a \in G</m>,
            there exists an <term>inverse element</term>
              <idx><h>Element</h><h>inverse</h></idx>
            in G, denoted by <m>a^{-1}</m>, such that
            <me>
              a \circ a^{-1} = a^{-1} \circ a = e
            </me>.
          </p>
        </li>
      </ul>
    </p>
    <p>
      A group <m>G</m> with the property that <m>a \circ b = b \circ a</m> for all
      <m>a,
      b \in G</m> is called <term>abelian</term>
          <idx><h>Abelian group</h></idx>
          <idx><h>Group</h><h>abelian</h></idx>
      or <term>commutative</term>.
          <idx><h>Group</h><h>commutative</h></idx>
      Groups not satisfying this property are said to be <term>nonabelian</term>
          <idx><h>Group</h><h>nonabelian</h></idx>
      or <term>noncommutative</term>.
          <idx><h>Group</h><h>noncommutative</h></idx>
    </p>
  </subsection>
  <subsection>
    <title>Examples of Groups</title>
    <example xml:id="example-groups-integers">
      <title>Group of All Integers</title>
      <p>
        The integers <m>{\mathbb Z } = \{ \ldots , -1, 0, 1, 2, \ldots \}</m> form a group under the operation of addition.
        The binary operation on two integers <m>m,
        n \in {\mathbb Z}</m> is just their sum.
        Since the integers under addition already have a well-established notation,
        we will use the operator <m>+</m> instead of <m>\circ</m>;
        that is, we shall write <m>m + n</m> instead of <m>m \circ n</m>.
        The identity is 0, and the inverse of
        <m>n \in {\mathbb Z}</m> is written as <m>-n</m> instead of <m>n^{-1}</m>.
        Notice that the set of integers under addition have the additional property that
        <m>m + n = n + m</m> and therefore form an abelian group.
      </p>
    </example>
    <p>
      Most of the time we will write <m>ab</m> instead of <m>a \circ b</m>;
      however, if the group already has a natural operation such as addition in the integers,
      we will use that operation.
      That is, if we are adding two integers,
      we still write <m>m + n</m>,
      <m>-n</m> for the inverse, and 0 for the identity as usual.
      We also write <m>m - n</m> instead of <m>m + (-n)</m>.
    </p>
    <p>
      It is often convenient to describe a group in terms of an addition or multiplication table.
      Such a table is called a <term>Cayley table</term>.
          <idx><h>Cayley table</h></idx>
    </p>
    <example xml:id="example-groups-z5">
      <title>Group of Integers Modulo <m>5</m></title>
      <p>
        The integers mod <m>n</m> form a group under addition modulo <m>n</m>.
        Consider <m>{\mathbb Z}_5</m>,
        consisting of the equivalence classes of the integers 0, 1, 2, 3, and 4.
        We define the group operation on <m>{\mathbb Z}_5</m> by modular addition.
        We write the binary operation on the group additively;
        that is, we write <m>m + n</m>.
        The element 0 is the identity of the group and each element in <m>{\mathbb Z}_5</m> has an inverse.
        For instance, <m>2 + 3 = 3 + 2 = 0</m>.
        <xref ref="figure-z5-cayley" /> is a Cayley table for
        <m>{\mathbb Z}_5</m> (Contributed by <xref ref="rbeezer" />).
        By <xref ref="proposition-zn-equiv-classes" />,
        <m>{\mathbb Z}_n = \{0, 1, \ldots,
        n-1 \}</m> is a group under the binary operation of addition mod <m>n</m>.
      </p>
      <figure xml:id="figure-z5-cayley">
        <caption>Cayley table for <m>({\mathbb Z_5}, +)</m></caption>
        <p>
          <me>
            \begin{array}{c|ccccc} + &amp; 0 &amp; 1 &amp; 2 &amp; 3 &amp; 4 \\ \hline 0 &amp; 0 &amp; 1 &amp; 2 &amp; 3 &amp; 4 \\ 1 &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 0 \\ 2 &amp; 2 &amp; 3 &amp; 4 &amp; 0 &amp; 1 \\ 3 &amp; 3 &amp; 4 &amp; 0 &amp; 1 &amp; 2 \\ 4 &amp; 4 &amp; 0 &amp; 1 &amp; 2 &amp; 3 \end{array}
          </me>
        </p>
      </figure>
    </example>
    <example xml:id="example-groups-z6-mult">
      <title>Multiplication of Integers Modulo <m>6</m></title>
      <p>
        Not every set with a binary operation is a group.
        For example,
        if we let modular multiplication be the binary operation on <m>{\mathbb Z}_n</m>,
        then <m>{\mathbb Z}_n</m> fails to be a group.
        The element 1 acts as a group identity since
        <m>1 \cdot k = k \cdot 1 = k</m> for any <m>k \in {\mathbb Z}_n</m>;
        however, a multiplicative inverse for <m>0</m> does not exist since
        <m>0 \cdot k = k \cdot 0 = 0</m> for every <m>k</m> in <m>{\mathbb Z}_n</m>.
        Even if we consider the set <m>{\mathbb Z}_n \setminus \{0 \}</m>,
        we still may not have a group.
        For instance, let <m>2 \in {\mathbb Z}_6</m>.
        Then 2 has no multiplicative inverse since
        <md>
          <mrow>0 \cdot 2 &amp; = 0 \qquad 1 \cdot 2 = 2</mrow>
          <mrow>2 \cdot 2 &amp; = 4 \qquad 3 \cdot 2 = 0</mrow>
          <mrow>4 \cdot 2 &amp; = 2 \qquad 5 \cdot 2 = 4</mrow>
        </md>.
      </p>
      <p>
        By <xref ref="proposition-zn-equiv-classes" />,
        every nonzero <m>k</m> does have an inverse in
        <m>{\mathbb Z}_n</m> if <m>k</m> is relatively prime to <m>n</m>.
        Denote the set of all such nonzero elements in <m>{\mathbb Z}_n</m> by <m>U(n)</m>.
        <notation>
          <usage><m>U(n)</m></usage>
          <description>group of units in <m>\mathbb Z_n</m></description>
        </notation>
        Then <m>U(n)</m> is a group called the
        <term>group of units</term>
            <idx><h>Group</h><h>of units</h></idx>
        of <m>{\mathbb Z}_n</m>.
        <xref ref="figure-cayley-u8" /> is a Cayley table for the group <m>U(8)</m>.
      </p>
      <figure xml:id="figure-cayley-u8">
        <caption>Multiplication table for <m>U(8)</m></caption>
        <p>
          <me>
            \begin{array}{c|cccc} \cdot &amp; 1 &amp; 3 &amp; 5 &amp; 7 \\ \hline 1       &amp; 1 &amp; 3 &amp; 5 &amp; 7 \\ 3       &amp; 3 &amp; 1 &amp; 7 &amp; 5 \\ 5       &amp; 5 &amp; 7 &amp; 1 &amp; 3 \\ 7       &amp; 7 &amp; 5 &amp; 3 &amp; 1 \end{array}
          </me>
        </p>
      </figure>
    </example>
    <example xml:id="example-groups-nonabelian">
      <title>Symmetries of a Triangle is not Abelian</title>
      <p>
        The symmetries of an equilateral triangle described in <xref ref="section-mod-n-sym" /> form a nonabelian group.
        As we observed, it is not necessarily true that
        <m>\alpha \beta = \beta \alpha</m> for two symmetries <m>\alpha</m> and <m>\beta</m>.
        Using <xref ref="figure-s3" />, which is a Cayley table for this group,
        we can easily check that the symmetries of an equilateral triangle are indeed a group.
        We will denote this group by either <m>S_3</m> or <m>D_3</m>,
        for reasons that will be explained later.
      </p>
    </example>
    <example xml:id="example-groups-gl2">
      <title>Matrix Multiplication of <m>2\times 2</m> Matrices is a Group</title>
      <p>
        We use  <m>{\mathbb M}_2 ( {\mathbb R})</m> to denote the set of all <m>2 \times 2</m> matrices.
        Let <m>GL_2({\mathbb R})</m> be the subset of
        <m>{\mathbb M}_2 ( {\mathbb R})</m> consisting of invertible matrices;
        that is, a matrix
        <notation>
          <usage><m>\mathbb M_n(\mathbb R)</m></usage>
          <description>the <m>n \times n</m> matrices with entries in <m>\mathbb R</m></description>
        </notation>
        <me>
          A = \begin{pmatrix} a &amp; b \\ c &amp; d \end{pmatrix}
        </me>
        is in  <m>GL_2( {\mathbb R})</m> if there exists a matrix <m>A^{-1}</m> such that <m>A A^{-1} = A^{-1} A = I</m>,
        where <m>I</m> is the <m>2 \times 2</m> identity matrix.
        For <m>A</m> to have an inverse is equivalent to requiring that the determinant of <m>A</m> be nonzero;
        that is, <m>\det A = ad - bc \neq 0</m>.
        <notation>
          <usage><m>\det A</m></usage>
          <description>the determinant of <m>A</m></description>
        </notation>
        The set of invertible matrices forms a group called the
        <term>general linear group</term>.
            <idx><h>Group</h><h>general linear</h></idx>
        <notation>
          <usage><m>GL_n(\mathbb R)</m></usage>
          <description>the general linear group</description>
        </notation>
        The identity of the group is the identity matrix
        <me>
          I = \begin{pmatrix} 1 &amp; 0 \\ 0 &amp; 1 \end{pmatrix}
        </me>.
        The inverse of <m>A \in GL_2( {\mathbb R})</m> is
        <me>
          A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d &amp; -b \\ -c &amp; a \end{pmatrix}
        </me>.
        The product of two invertible matrices is again invertible.
        Matrix multiplication is associative,
        satisfying the other group axiom.
        For matrices it is not true in general that <m>AB = BA</m>;
        hence, <m>GL_2({\mathbb R})</m> is another example of a nonabelian group.
      </p>
    </example>

    <example xml:id="example-groups-quaterions">
      <title>Group of Quaternions</title>
      <p>
        Let
        <md>
          <mrow>1 &amp; = \begin{pmatrix} 1 &amp; 0\\ 0 &amp; 1 \end{pmatrix} \qquad I = \begin{pmatrix} 0 &amp; 1\\ -1 &amp; 0 \end{pmatrix}</mrow>
          <mrow>J &amp; = \begin{pmatrix} 0 &amp; i\\ i &amp; 0 \end{pmatrix} \qquad K = \begin{pmatrix} i &amp; 0\\ 0 &amp; -i \end{pmatrix}</mrow>
        </md>,
        where <m>i^2 = -1</m>.
        Then the relations <m>I^2 = J^2 = K^2 = -1</m>,
        <m>IJ=K</m>, <m>JK = I</m>, <m>KI = J</m>,
        <m>JI = -K</m>, <m>KJ = -I</m>, and <m>IK = -J</m> hold.
        The set <m>Q_8 = \{\pm 1, \pm I, \pm J, \pm K  \}</m> is a group called the
        <term>quaternion group</term>.
        Notice that <m>Q_8</m> is noncommutative.
        <notation>
          <usage><m>Q_8</m></usage>
          <description>the group of quaternions</description>
        </notation>
            <idx> <h>Group</h> <h>quaternion</h> </idx>
            <idx> <h>Quaternions</h> </idx>
      </p>
    </example>
    <example xml:id="example-groups-c-star">
      <title>Group of Nonzero Complex Numbers</title>
      <p>
        Let <m>{\mathbb C}^\ast</m>be the set of nonzero complex numbers.
        <notation>
          <usage><m>\mathbb C^*</m></usage>
          <description>the multiplicative group of complex numbers</description>
        </notation>
        Under the operation of multiplication <m>{\mathbb C}^\ast</m> forms a group.
        The identity is 1.
        If <m>z = a+bi</m> is a nonzero complex number, then
        <me>
          z^{-1} = \frac{a -bi}{a^2 +b^2}
        </me>
        is the inverse of <m>z</m>.
        It is easy to see that the remaining group axioms hold.
      </p>
    </example>
    <p>
      A group is <term>finite</term>,
          <idx><h>Group</h><h>finite</h></idx>
      or has <term>finite order</term>,
      if it contains a finite number of elements;
      otherwise, the group is said to be <term>infinite</term>
      or to have <term>infinite order</term>.
          <idx><h>Group</h><h>infinite</h></idx>
      The <term>order</term>
          <idx><h>Group</h><h>order of</h></idx>
      of a finite group is the number of elements that it contains.
      If <m>G</m> is a group containing <m>n</m> elements, we write <m>|G| = n</m>.
      <notation>
        <usage><m>|G|</m></usage>
        <description>the order of a group</description>
      </notation>
      The group <m>{\mathbb Z}_5</m> is a finite group of order 5;
      the integers <m>{\mathbb Z}</m> form an infinite group under addition,
      and we sometimes write <m>|{\mathbb Z}| = \infty</m>.
    </p>
  </subsection>
  <subsection>
    <title>Basic Properties of Groups</title>
    <proposition>
      <statement>
        <p>
          The identity element in a group <m>G</m> is unique;
          that is, there exists only one element <m>e \in G</m> such that
          <m>eg = ge = g</m> for all <m>g \in G</m>.
        </p>
      </statement>
      <proof>
        <p>
          Suppose that <m>e</m> and <m>e'</m> are both identities in <m>G</m>.
          Then <m>eg = ge = g</m> and
          <m>e'g = ge' = g</m> for all <m>g \in G</m>.
          We need to show that <m>e = e'</m>.
          If we think of <m>e</m> as the identity, then <m>ee' = e'</m>;
          but if <m>e'</m> is the identity, then <m>ee' = e</m>.
          Combining these two equations,
          we have <m>e = ee' = e'</m>.
          (<xref ref="dfarmer" /> helped with this proof.)
        </p>
      </proof>
    </proposition>
    <p>
      Inverses in a group are also unique.
      If <m>g'</m> and <m>g''</m> are both inverses of an element <m>g</m> in a group <m>G</m>,
      then <m>gg' = g'g = e</m> and <m>gg'' = g''g = e</m>.
      We want to show that <m>g' = g''</m>,
      but <m>g' = g'e = g'(gg'') = (g'g)g'' = eg'' = g''</m>.
      We summarize this fact in the following proposition.
    </p>
    <proposition>
      <statement>
        <p>
          If <m>g</m> is any element in a group <m>G</m>,
          then the inverse of <m>g</m>,
          denoted by <m>g^{-1}</m>, is unique.
        </p>
      </statement>
    </proposition>
    <proposition xml:id="proposition-inverse-of-products">
      <statement>
        <p>
          Let <m>G</m> be a group.
          If <m>a, b \in G</m>, then <m>(ab)^{-1} = b^{-1}a^{-1}</m>.
        </p>
      </statement>
      <proof>
        <p>
          Let <m>a, b \in G</m>.
          Then <m>abb^{-1}a^{-1} = aea^{-1} = aa^{-1} = e</m>.
          Similarly, <m>b^{-1}a^{-1}ab = e</m>.
          But by the previous proposition, inverses are unique;
          hence, <m>(ab)^{-1} = b^{-1}a^{-1}</m>.
        </p>
      </proof>
    </proposition>
    <proposition>
      <statement>
        <p>
          Let <m>G</m> be a group.
          For any <m>a \in G</m>, <m>(a^{-1})^{-1} = a</m>.
        </p>
      </statement>
      <proof>
        <p>
          Observe that <m>a^{-1} (a^{-1})^{-1} = e</m>.
          Consequently,
          multiplying both sides of this equation by <m>a</m>, we have
          <me>
            (a^{-1})^{-1} = e (a^{-1})^{-1} = a a^{-1} (a^{-1})^{-1} = ae = a
          </me>.
        </p>
      </proof>
    </proposition>
    <p>
      It makes sense to write equations with group elements and group operations.
      If <m>a</m> and <m>b</m> are two elements in a group <m>G</m>,
      does there exist an element <m>x \in G</m> such that <m>ax = b</m>?
      If such an <m>x</m> does exist, is it unique?
      The following proposition answers both of these questions positively.
    </p>
    <proposition xml:id="proposition-group-equations">
      <statement>
        <p>
          Let <m>G</m> be a group and <m>a</m> and <m>b</m> be any two elements in <m>G</m>.
          Then the equations <m>ax = b</m> and <m>xa = b</m> have unique solutions in <m>G</m>.
        </p>
      </statement>
      <proof>
        <p>
          Suppose that <m>ax = b</m>.
          We must show that such an <m>x</m> exists.
          Multiplying both sides of <m>ax = b</m> by <m>a^{-1}</m>,
          we have <m>x = ex = a^{-1}ax = a^{-1}b</m>.
        </p>
        <p>
          To show uniqueness,
          suppose that <m>x_1</m> and <m>x_2</m> are both solutions of <m>ax = b</m>;
          then <m>ax_1 = b = ax_2</m>.
          So <m>x_1 = a^{-1}ax_1 = a^{-1}ax_2 = x_2</m>.
          The proof for the existence and uniqueness of the solution of <m>xa = b</m> is similar.
        </p>
      </proof>
    </proposition>
    <proposition>
      <statement>
        <p>
          If <m>G</m> is a group and <m>a, b, c \in G</m>,
          then <m>ba = ca</m> implies <m>b = c</m> and <m>ab = ac</m> implies <m>b = c</m>.
        </p>
      </statement>
    </proposition>
    <p>
      This proposition tells us that the
      <term>right and left cancellation laws</term><idx><h>Cancellation law</h><h>for groups</h></idx> are true in groups.
      We leave the proof as an exercise.
    </p>
    <p>
      We can use exponential notation for groups just as we do in ordinary algebra.
      If <m>G</m> is a group and <m>g \in G</m>,
      then we define <m>g^0 = e</m>.
      For <m>n \in {\mathbb N}</m>, we define
      <me>
        g^n = \underbrace{g \cdot g \cdots g}_{n \; \text{times}}
      </me>
      and
      <me>
        g^{-n} = \underbrace{g^{-1} \cdot g^{-1} \cdots g^{-1}}_{n \; \text{times}}
      </me>.
    </p>
    <theorem xml:id="theorem-exponent-laws">
      <statement>
        <p>
          In a group, the usual laws of exponents hold;
          that is, for all <m>g, h \in G</m>,
          <ol>
            <li>
              <p>
                <m>g^mg^n = g^{m+n}</m> for all <m>m, n \in {\mathbb Z}</m>;
              </p>
            </li>
            <li>
              <p>
                <m>(g^m)^n = g^{mn}</m> for all <m>m, n \in {\mathbb Z}</m>;
              </p>
            </li>
            <li>
              <p>
                <m>(gh)^n = (h^{-1}g^{-1})^{-n}</m> for all <m>n \in {\mathbb Z}</m>.
                Furthermore, if <m>G</m> is abelian,
                then <m>(gh)^n = g^nh^n</m>.
              </p>
            </li>
          </ol>
        </p>
      </statement>
    </theorem>
    <p>
      We will leave the proof of this theorem as an exercise.
      Notice that <m>(gh)^n \neq g^nh^n</m> in general,
      since the group may not be abelian.
      If the group is <m>{\mathbb Z}</m> or <m>{\mathbb Z}_n</m>,
      we write the group operation additively and the exponential operation multiplicatively;
      that is, we write <m>ng</m> instead of <m>g^n</m>.
      The laws of exponents now become
      <ol>
        <li>
          <p>
            <m>mg + ng = (m+n)g</m> for all <m>m, n \in {\mathbb Z}</m>;
          </p>
        </li>
        <li>
          <p>
            <m>m(ng) = (mn)g</m> for all <m>m, n \in {\mathbb Z}</m>;
          </p>
        </li>
        <li>
          <p>
            <m>m(g + h) = mg + mh</m> for all <m>n \in {\mathbb Z}</m>.
          </p>
        </li>
      </ol>
    </p>
    <p>
      It is important to realize that the last statement can be made only because <m>{\mathbb Z}</m> and
      <m>{\mathbb Z}_n</m> are commutative groups.
    </p>
  </subsection>
  <subsection>
    <title>Historical Note</title>
    <p>
      Although the first clear axiomatic definition of a group was not given until the late 1800s,
      group-theoretic methods had been employed before this time in the development of many areas of mathematics,
      including geometry and the theory of algebraic equations.
    </p>
    <p>
      Joseph-Louis Lagrange
          <idx><h>Lagrange, Joseph-Louis</h></idx>
      used group-theoretic methods in a 1770<ndash />1771 memoir to study methods of solving polynomial equations.
      Later, Évariste Galois
          <idx><h>Galois, Évariste</h></idx>
      (1811<ndash />1832) succeeded in developing the mathematics necessary to determine exactly which polynomial equations could be solved in terms of the polynomial's coefficients.
      Galois' primary tool was group theory.
    </p>
    <p>
      The study of geometry was revolutionized in 1872 when Felix Klein
          <idx><h>Klein, Felix</h></idx>
      proposed that geometric spaces should be studied by examining those properties that are invariant under a transformation of the space.
      Sophus Lie,
          <idx><h>Lie, Sophus</h></idx>
      a contemporary of Klein,
      used group theory to study solutions of partial differential equations.
      One of the first modern treatments of group theory appeared in William Burnside's
          <idx><h>Burnside, William</h></idx>
      <pubtitle>The Theory of Groups of Finite Order</pubtitle>
      [1], first published in 1897.
    </p>
  </subsection>
</section>