Abstract Algebra: Theory and Applications, Remixed!

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.

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 \(n\text{.}\)

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

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 \(U(8)\text{.}\)

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

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

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

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.

Let

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

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