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Section 2.2 Definitions and Examples

The integers mod \(n\) and the symmetries of a triangle or a rectangle are examples of groups. A binary operation or law of composition on a set \(G\) is a function \(G \times G \rightarrow G\) that assigns to each pair \((a,b) \in G \times G\) a unique element \(a \circ b\text{,}\) or \(ab\) in \(G\text{,}\) called the composition of \(a\) and \(b\text{.}\) A group \((G, \circ )\) is a set \(G\) together with a law of composition \((a,b) \mapsto a \circ b\) that satisfies the following axioms.

  • The law of composition is associative. That is,

    \begin{equation*} (a \circ b) \circ c = a \circ (b \circ c) \end{equation*}

    for \(a, b, c \in G\text{.}\)

  • There exists an element \(e \in G\text{,}\) called the identity element, such that for any element \(a \in G\)

    \begin{equation*} e \circ a = a \circ e = a\text{.} \end{equation*}
  • For each element \(a \in G\text{,}\) there exists an inverse element in G, denoted by \(a^{-1}\text{,}\) such that

    \begin{equation*} a \circ a^{-1} = a^{-1} \circ a = e\text{.} \end{equation*}

A group \(G\) with the property that \(a \circ b = b \circ a\) for all \(a, b \in G\) is called abelian or commutative. Groups not satisfying this property are said to be nonabelian or noncommutative.

Example 2.8.

The integers \({\mathbb Z } = \{ \ldots , -1, 0, 1, 2, \ldots \}\) form a group under the operation of addition. The binary operation on two integers \(m, n \in {\mathbb Z}\) is just their sum. Since the integers under addition already have a well-established notation, we will use the operator \(+\) instead of \(\circ\text{;}\) that is, we shall write \(m + n\) instead of \(m \circ n\text{.}\) The identity is \(0\text{,}\) and the inverse of \(n \in {\mathbb Z}\) is written as \(-n\) instead of \(n^{-1}\text{.}\) Notice that the set of integers under addition have the additional property that \(m + n = n + m\) and therefore form an abelian group.

Most of the time we will write \(ab\) instead of \(a \circ b\text{;}\) 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 \(m + n\text{,}\) \(-n\) for the inverse, and 0 for the identity as usual. We also write \(m - n\) instead of \(m + (-n)\text{.}\)

It is often convenient to describe a group in terms of an addition or multiplication table. Such a table is called a Cayley table.

Example 2.9.

The integers mod \(n\) form a group under addition modulo \(n\text{.}\) Consider \({\mathbb Z}_5\text{,}\) consisting of the equivalence classes of the integers \(0\text{,}\) \(1\text{,}\) \(2\text{,}\) \(3\text{,}\) and \(4\text{.}\) We define the group operation on \({\mathbb Z}_5\) by modular addition. We write the binary operation on the group additively; that is, we write \(m + n\text{.}\) The element 0 is the identity of the group and each element in \({\mathbb Z}_5\) has an inverse. For instance, \(2 + 3 = 3 + 2 = 0\text{.}\) Figure 2.10 is a Cayley table for \({\mathbb Z}_5\text{.}\) By Proposition 2.4, \({\mathbb Z}_n = \{0, 1, \ldots, n-1 \}\) is a group under the binary operation of addition mod \(n\text{.}\)

\begin{equation*} \begin{array}{c|ccccc} + & 0 & 1 & 2 & 3 & 4 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 \\ 1 & 1 & 2 & 3 & 4 & 0 \\ 2 & 2 & 3 & 4 & 0 & 1 \\ 3 & 3 & 4 & 0 & 1 & 2 \\ 4 & 4 & 0 & 1 & 2 & 3 \end{array} \end{equation*}
Figure 2.10. Cayley table for \(({\mathbb Z_5}, +)\)
Example 2.11.

Not every set with a binary operation is a group. For example, if we let modular multiplication be the binary operation on \({\mathbb Z}_n\text{,}\) then \({\mathbb Z}_n\) fails to be a group. The element 1 acts as a group identity since \(1 \cdot k = k \cdot 1 = k\) for any \(k \in {\mathbb Z}_n\text{;}\) however, a multiplicative inverse for \(0\) does not exist since \(0 \cdot k = k \cdot 0 = 0\) for every \(k\) in \({\mathbb Z}_n\text{.}\) Even if we consider the set \({\mathbb Z}_n \setminus \{0 \}\text{,}\) we still may not have a group. For instance, let \(2 \in {\mathbb Z}_6\text{.}\) Then 2 has no multiplicative inverse since

\begin{align*} 0 \cdot 2 & = 0 \qquad 1 \cdot 2 = 2\\ 2 \cdot 2 & = 4 \qquad 3 \cdot 2 = 0\\ 4 \cdot 2 & = 2 \qquad 5 \cdot 2 = 4\text{.} \end{align*}

By Proposition 2.4, every nonzero \(k\) does have an inverse in \({\mathbb Z}_n\) if \(k\) is relatively prime to \(n\text{.}\) Denote the set of all such nonzero elements in \({\mathbb Z}_n\) by \(U(n)\text{.}\) Then \(U(n)\) is a group called the group of units of \({\mathbb Z}_n\text{.}\) Figure 2.12 is a Cayley table for the group \(U(8)\text{.}\)

\begin{equation*} \begin{array}{c|cccc} \cdot & 1 & 3 & 5 & 7 \\ \hline 1 & 1 & 3 & 5 & 7 \\ 3 & 3 & 1 & 7 & 5 \\ 5 & 5 & 7 & 1 & 3 \\ 7 & 7 & 5 & 3 & 1 \end{array} \end{equation*}
Figure 2.12. Multiplication table for \(U(8)\)
Example 2.13.

The symmetries of an equilateral triangle described in Section 2.1 form a nonabelian group. As we observed, it is not necessarily true that \(\alpha \beta = \beta \alpha\) for two symmetries \(\alpha\) and \(\beta\text{.}\) Using Figure 2.7, 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 \(S_3\) or \(D_3\text{,}\) for reasons that will be explained later.

Example 2.14.

We use \({\mathbb M}_2 ( {\mathbb R})\) to denote the set of all \(2 \times 2\) matrices. Let \(GL_2({\mathbb R})\) be the subset of \({\mathbb M}_2 ( {\mathbb R})\) consisting of invertible matrices; that is, a matrix

\begin{equation*} A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \end{equation*}

is in \(GL_2( {\mathbb R})\) if there exists a matrix \(A^{-1}\) such that \(A A^{-1} = A^{-1} A = I\text{,}\) where \(I\) is the \(2 \times 2\) identity matrix. For \(A\) to have an inverse is equivalent to requiring that the determinant of \(A\) be nonzero; that is, \(\det A = ad - bc \neq 0\text{.}\) The set of invertible matrices forms a group called the general linear group. The identity of the group is the identity matrix

\begin{equation*} I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\text{.} \end{equation*}

The inverse of \(A \in GL_2( {\mathbb R})\) is

\begin{equation*} A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\text{.} \end{equation*}

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 \(AB = BA\text{;}\) hence, \(GL_2({\mathbb R})\) is another example of a nonabelian group.

Example 2.15.


\begin{align*} 1 & = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \qquad I = \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}\\ J & = \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \qquad K = \begin{pmatrix} i & 0\\ 0 & -i \end{pmatrix}\text{,} \end{align*}

where \(i^2 = -1\text{.}\) Then the relations \(I^2 = J^2 = K^2 = -1\text{,}\) \(IJ=K\text{,}\) \(JK = I\text{,}\) \(KI = J\text{,}\) \(JI = -K\text{,}\) \(KJ = -I\text{,}\) and \(IK = -J\) hold. The set \(Q_8 = \{\pm 1, \pm I, \pm J, \pm K \}\) is a group called the quaternion group. Notice that \(Q_8\) is noncommutative.

Example 2.16.

Let \({\mathbb C}^\ast\) be the set of nonzero complex numbers. Under the operation of multiplication \({\mathbb C}^\ast\) forms a group. The identity is \(1\text{.}\) If \(z = a+bi\) is a nonzero complex number, then

\begin{equation*} z^{-1} = \frac{a -bi}{a^2 +b^2} \end{equation*}

is the inverse of \(z\text{.}\) It is easy to see that the remaining group axioms hold.

A group is finite, or has finite order, if it contains a finite number of elements; otherwise, the group is said to be infinite or to have infinite order. The order of a finite group is the number of elements that it contains. If \(G\) is a group containing \(n\) elements, we write \(|G| = n\text{.}\) The group \({\mathbb Z}_5\) is a finite group of order \(5\text{;}\) the integers \({\mathbb Z}\) form an infinite group under addition, and we sometimes write \(|{\mathbb Z}| = \infty\text{.}\)

Subsection 2.2.1 Historical Note

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.

Joseph-Louis Lagrange used group-theoretic methods in a 1770–1771 memoir to study methods of solving polynomial equations. Later, Évariste Galois (1811–1832) succeeded in developing the mathematics necessary to determine exactly which polynomial equations could be solved in terms of the coefficients of the polynomial. Galois' primary tool was group theory.

The study of geometry was revolutionized in 1872 when Felix Klein proposed that geometric spaces should be studied by examining those properties that are invariant under a transformation of the space. Sophus Lie, 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 The Theory of Groups of Finite Order [1], first published in 1897.

Reading Questions 2.2.2 Reading Questions


What does it mean for a group to be abelian? Be as specific as possible.


What is the inverse of the element 5 in (a) the group \(\Z_8\) and (b) the group \(U(8)\text{?}\) Breifly explain your answers.


What is the order of the group \(U(8)\text{?}\) Explain.


After reading the section, what questions do you still have? Write at least one well formulated question (even if you think you understand everything).

Exercises 2.2.3 Practice Problems


Find all \(x \in {\mathbb Z}\) satisfying each of the following equations.

  1. \(\displaystyle 3x \equiv 2 \pmod{7}\)

  2. \(\displaystyle 5x + 1 \equiv 13 \pmod{23}\)

  3. \(\displaystyle 5x + 1 \equiv 13 \pmod{26}\)

  4. \(\displaystyle 9x \equiv 3 \pmod{5}\)

  5. \(\displaystyle 5x \equiv 1 \pmod{6}\)

  6. \(\displaystyle 3x \equiv 1 \pmod{6}\)


(a) \(3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}\text{;}\) (c) \(18 + 26 \mathbb Z\text{;}\) (e) \(5 + 6 \mathbb Z\text{.}\)


Which of the following multiplication tables defined on the set \(G = \{ a, b, c, d \}\) form a group? Support your answer in each case.

  1. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & c & d & a \\ b & b & b & c & d \\ c & c & d & a & b \\ d & d & a & b & c \end{array} \end{equation*}
  2. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & d & c \\ c & c & d & a & b \\ d & d & c & b & a \end{array} \end{equation*}
  3. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & c & d & a \\ c & c & d & a & b \\ d & d & a & b & c \end{array} \end{equation*}
  4. \begin{equation*} \begin{array}{c|cccc} \circ & a & b & c & d \\ \hline a & a & b & c & d \\ b & b & a & c & d \\ c & c & b & a & d \\ d & d & d & b & c \end{array} \end{equation*}

(a) Not a group; (c) a group.


Write out Cayley tables for groups formed by the symmetries of a rectangle and for \(({\mathbb Z}_4, +)\text{.}\) How many elements are in each group? Are the groups the same? Why or why not?


Both groups have order 4, but they are not the same.


Describe the symmetries of a rhombus and prove that the set of symmetries forms a group. Give Cayley tables for both the symmetries of a rectangle and the symmetries of a rhombus. Are the symmetries of a rectangle and those of a rhombus the same?


Give a multiplication table for the group \(U(12)\text{.}\)

\begin{equation*} \begin{array}{c|cccc} \cdot & 1 & 5 & 7 & 11 \\ \hline 1 & 1 & 5 & 7 & 11 \\ 5 & 5 & 1 & 11 & 7 \\ 7 & 7 & 11 & 1 & 5 \\ 11 & 11 & 7 & 5 & 1 \end{array} \end{equation*}

Consider \((\Z,-)\text{,}\) the integers with operation subtraction. Explain why this is not a group. Say specifically what goes wrong.


It doesn't matter that subtraction is not commutative. The other three properties do not hold though. Can you say why?


Consider the operation that gives the distance between two numbers on the number line. In other words, define \(\ast\) on \(\R\) by \(x \ast y = |x-y|\text{.}\) Analyze this operation. That is, decide whether the operation is associative, commutative, has an identity, and has inverses. Is \((\R, \ast)\) a group?


Consider the set \(S = \{A,B,C\}\text{.}\) It turns out that there are \(19683\) different operations on this set. Below, you are asked to describe five of them, each with some given properties. To do this, simply write out the \(3\times 3\) operation table. Then say how you know that the operation you gave has the perperties requested.

  1. \(\ast\) is not commutative.

  2. \(\ast\) is not associative, but is commutative.
  3. \(S\) has an identity with respect to \(\ast\text{,}\) but the identity is not \(A\text{.}\)
  4. \(S\) has an identity, but not every element has an inverse under \(\ast\text{.}\)
  5. \(\ast\) is commutative, \(S\) has an identity, and every element has an inverse under \(\ast\text{.}\)

Prove that every group of order 3 is abelian. See part (e) of the previous problem.


You might as well use \(A\) for the identity. This takes care of one row and one column. There are only 4 spots left in the Cayley table.


Define \(\ast\) on the real numbers \(\R\) by \(x\ast y = x+y-5\text{.}\) Is \((\R, \ast)\) a group? Is it an abelian group?

Exercises 2.2.4 Collected Homework


Let \(S = {\mathbb R} \setminus \{ -1 \}\) and define a binary operation on \(S\) by \(a \ast b = a + b + ab\text{.}\) Prove that \((S, \ast)\) is an abelian group.


Careful here for inverses. They are not the usual inverses. Say what they are and prove you are correct.


We know that a set \(S\) might or might not have an identity element with respect to the operation \(\ast\text{.}\) Is it possible for a set to have more than one identity element? Explain what this would mean in terms of the Cayley (operation) table, and why this answers the question.