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Abstract Algebra: Theory and Applications

Thomas W. Judson

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Section 3.2 Definitions and Examples

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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×G→G that assigns to each pair (a,b)∈G×G a unique element a∘b, or ab in G, called the composition of a and b. A group (G,∘) is a set G together with a law of composition (a,b)↦a∘b that satisfies the following axioms.
  • The law of composition is associative. That is,
    (a∘b)∘c=a∘(b∘c)
    for a,b,c∈G.
  • There exists an element e∈G, called the identity element, such that for any element a∈G
    e∘a=a∘e=a.
  • For each element a∈G, there exists an inverse element in G, denoted by a−1, such that
    a∘a−1=a−1∘a=e.
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A group G with the property that a∘b=b∘a for all a,b∈G is called abelian or commutative. Groups not satisfying this property are said to be nonabelian or noncommutative.
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Example 3.8.

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The integers Z={…,−1,0,1,2,…} form a group under the operation of addition. The binary operation on two integers m,n∈Z is just their sum. Since the integers under addition already have a well-established notation, we will use the operator + instead of ∘; that is, we shall write m+n instead of m∘n. The identity is 0, and the inverse of n∈Z is written as −n instead of n−1. Notice that the set of integers under addition have the additional property that m+n=n+m and therefore form an abelian group.
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Most of the time we will write ab instead of a∘b; 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, −n for the inverse, and 0 for the identity as usual. We also write m−n instead of m+(−n).
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It is often convenient to describe a group in terms of an addition or multiplication table. Such a table is called a Cayley table.
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Example 3.9.

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The integers mod n form a group under addition modulo n. Consider Z5, consisting of the equivalence classes of the integers 0, 1, 2, 3, and 4. We define the group operation on Z5 by modular addition. We write the binary operation on the group additively; that is, we write m+n. The element 0 is the identity of the group and each element in Z5 has an inverse. For instance, 2+3=3+2=0. Figure 3.10 is a Cayley table for Z5. By Proposition 3.4, Zn={0,1,…,n−1} is a group under the binary operation of addition mod n.
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+01234001234112340223401334012440123
Figure 3.10. Cayley table for (Z5,+)
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Example 3.11.

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Not every set with a binary operation is a group. For example, if we let modular multiplication be the binary operation on Zn, then Zn fails to be a group. The element 1 acts as a group identity since 1⋅k=k⋅1=k for any k∈Zn; however, a multiplicative inverse for 0 does not exist since 0⋅k=k⋅0=0 for every k in Zn. Even if we consider the set Zn∖{0}, we still may not have a group. For instance, let 2∈Z6. Then 2 has no multiplicative inverse since
0⋅2=01⋅2=22⋅2=43⋅2=04⋅2=25⋅2=4.
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By Proposition 3.4, every nonzero k does have an inverse in Zn if k is relatively prime to n. Denote the set of all such nonzero elements in Zn by U(n). Then U(n) is a group called the group of units of Zn. Figure 3.12 is a Cayley table for the group U(8).
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⋅135711357331755571377531
Figure 3.12. Multiplication table for U(8)
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Example 3.13.

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The symmetries of an equilateral triangle described in Section 3.1 form a nonabelian group. As we observed, it is not necessarily true that αβ=βα for two symmetries α and β. Using Figure 3.7, 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 S3 or D3, for reasons that will be explained later.
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Example 3.14.

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We use M2(R) to denote the set of all 2×2 matrices. Let GL2(R) be the subset of M2(R) consisting of invertible matrices; that is, a matrix
A=(abcd)
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is in GL2(R) if there exists a matrix A−1 such that AA−1=A−1A=I, where I is the 2×2 identity matrix. For A to have an inverse is equivalent to requiring that the determinant of A be nonzero; that is, detA=ad−bc≠0. The set of invertible matrices forms a group called the general linear group. The identity of the group is the identity matrix
I=(1001).
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The inverse of A∈GL2(R) is
A−1=1ad−bc(d−b−ca).
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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; hence, GL2(R) is another example of a nonabelian group.
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Example 3.15.

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Let
1=(1001)I=(01−10)J=(0ii0)K=(i00−i),
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where i2=−1. Then the relations I2=J2=K2=−1, IJ=K, JK=I, KI=J, JI=−K, KJ=−I, and IK=−J hold. The set Q8={±1,±I,±J,±K} is a group called the quaternion group. Notice that Q8 is noncommutative.
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Example 3.16.

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Let C∗ be the set of nonzero complex numbers. Under the operation of multiplication C∗ forms a group. The identity is 1. If z=a+bi is a nonzero complex number, then
z−1=a−bia2+b2
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is the inverse of z. It is easy to see that the remaining group axioms hold.
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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. The group Z5 is a finite group of order 5; the integers Z form an infinite group under addition, and we sometimes write |Z|=∞.
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Subsection Basic Properties of Groups

Proof.

Suppose that e and e′ are both identities in G. Then eg=ge=g and e′g=ge′=g for all g∈G. We need to show that e=e′. If we think of e as the identity, then ee′=e′; but if e′ is the identity, then ee′=e. Combining these two equations, we have e=ee′=e′.
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Inverses in a group are also unique. If g′ and g″ are both inverses of an element g in a group G, then gg′=g′g=e and gg″=g″g=e. We want to show that g′=g″, but g′=g′e=g′(gg″)=(g′g)g″=eg″=g″. We summarize this fact in the following proposition.

Proof.

Let a,b∈G. Then abb−1a−1=aea−1=aa−1=e. Similarly, b−1a−1ab=e. But by the previous proposition, inverses are unique; hence, (ab)−1=b−1a−1.

Proof.

Observe that a−1(a−1)−1=e. Consequently, multiplying both sides of this equation by a, we have
(a−1)−1=e(a−1)−1=aa−1(a−1)−1=ae=a.
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It makes sense to write equations with group elements and group operations. If a and b are two elements in a group G, does there exist an element x∈G such that ax=b? If such an x does exist, is it unique? The following proposition answers both of these questions positively.

Proof.

Suppose that ax=b. We must show that such an x exists. We can multiply both sides of ax=b by a−1 to find x=ex=a−1ax=a−1b.
To show uniqueness, suppose that x1 and x2 are both solutions of ax=b; then ax1=b=ax2. So x1=a−1ax1=a−1ax2=x2. The proof for the existence and uniqueness of the solution of xa=b is similar.
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This proposition tells us that the right and left cancellation laws are true in groups. We leave the proof as an exercise.
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We can use exponential notation for groups just as we do in ordinary algebra. If G is a group and g∈G, then we define g0=e. For n∈N, we define
gn=g⋅g⋯g⏟ntimes
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and
g−n=g−1⋅g−1⋯g−1⏟ntimes.
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We will leave the proof of this theorem as an exercise. Notice that (gh)n≠gnhn in general, since the group may not be abelian. If the group is Z or Zn, we write the group operation additively and the exponential operation multiplicatively; that is, we write ng instead of gn. The laws of exponents now become
  1. mg+ng=(m+n)g for all m,n∈Z;
  2. m(ng)=(mn)g for all m,n∈Z;
  3. m(g+h)=mg+mh for all n∈Z.
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It is important to realize that the last statement can be made only because Z and Zn are commutative groups.
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Subsection Historical Note

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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.
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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.
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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.
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