We show that \(\Z_7\) with \(\oplus\) satisfies the properties of a group from Definition 14.2. As the remainder of division by \(7\) is always in \(\Z_7\) we have that \(\oplus\) is indeed a binary operation on \(\Z_7\text{.}\)
Identity: Because \(a\oplus 0=(a+0)\fmod 7=a\) and \(0\oplus a = (a + 0) \fmod 7 = a\) for all \(a\in\Z_7\text{,}\)\(0\) is the identity element of \(\oplus\text{.}\)
Inverse: We have \(0 \oplus 0 = (0 + 0) \fmod 7 = 0\text{.}\) So \(0\) is the identity element in \(\Z_7\text{.}\) Let \(a\in\Z_7\) with \(a \neq 0\) and let \(b=7-a\text{.}\) Then,
Associativity: Let \(a\in\Z_7\text{,}\)\(b\in\Z_7\text{,}\) and \(c\in\Z_7\text{.}\) By Theorem 3.46 we only need to show that \((a+(b+c))\fmod 7=((a+b)+c)\fmod 7\text{.}\) This holds since \(a+(b+c)=(a+b)+c\) for all integers \(a\text{,}\)\(b\text{,}\) and \(c\) by the associative property of the integers. Hence \(\oplus\) is associative.
Commutativity: By the commutative property of the integers we have \(a+b=b+a\) for all integers \(a\) and \(b\text{.}\) Thus also for all \(a\in\Z_7\) and \(b\in\Z_7\text{,}\) we have \(a+b=b+a\) and \(a\oplus b=(a+b)\fmod 7=(b+a)\fmod 7=b\oplus a\text{.}\) We can also deduce the commutativity of \(\oplus\) from the symmetry of the addition table in Table 14.15.
In general we have that for any natural number \(n\) in \((\Z_n,\oplus)\) where \(a\oplus b=(a+b)\fmod m\) is a group. We give an overview over this result in the video in Figure 14.21 and then go through the result and its proof in detail below.
Let \(n\in\N\text{.}\) The set \(\Z_n=\{0,1,2,\dots,n-1\}\) with the operation \(\oplus:\Z_n\times\Z_n\to\Z_n\text{,}\)\(a\oplus b=(a+b)\fmod n\) is a group.
Identity: Let \(a\in\Z_n\text{.}\) We have \(a\oplus 0=(a+0)\fmod n=a\fmod n=a\) and similarly \(0\oplus a=(0+a)\fmod n=a\fmod n=a\text{.}\) Hence \(0\) is an identity element with respect to \(\oplus\text{.}\)
Inverses: We have \(0\oplus 0=(0+0)\fmod n=0\text{.}\) Thus 0 is the inverse of 0 in \(\Z_n\) with respect to \(\oplus\text{.}\) Now consider \(a\in\Z_n\) and \(a \neq 0\text{.}\) Let \(b=n-a\text{.}\) So \(b \in \Z_n\text{.}\) Then
\begin{equation*}
a\oplus b=a\oplus(n-a)=(a+(n-a)) \fmod n =(a-a+n)\fmod n = 0\fmod n\text{.}
\end{equation*}
Associativity: The associativity of \(\oplus\) follows from the associativity of \(+\text{.}\) Let \(a\in\Z_n\text{,}\)\(b\in\Z_n\text{,}\) and \(c\in\Z_n\text{.}\) By Theorem 3.46 we only need to show that \((a+(b+c))\fmod n=((a+b)+c)\fmod n\text{.}\) This holds since \(a+(b+c)=(a+b)+c\) for all integers \(a\text{,}\)\(b\text{,}\) and \(c\) by the associative property of the integers. Hence \(\oplus\) is associative.
Commutativity: By the commutative property of the integers we have \(a+b=b+a\) for all integers \(a\) and \(b\text{.}\) Thus also for all \(a\in\Z_n\) and \(b\in\Z_n\) we have \(a+b=b+a\) and \(a\oplus b=(a+b)\fmod n=(b+a)\fmod n=b\oplus a\text{.}\)
Directly from the proof of Theorem 14.22Item 2 we obtain a method for finding inverses in \((\Z_n,\oplus)\text{.}\) Namely if \(a\in\Z_n\) and \(a \neq 0\) then \(b=n-a\in\Z_n\) and \(a\oplus b=0\text{.}\)
We have \(5\oplus 7=(5+7)\fmod 12=12\fmod 12=0\text{.}\) As the group \((\Z_{12},\oplus)\) is commutative this shows that 7 is the inverse of \(5\text{.}\)
Let \(={ 0,1,2,\dots, 60}\text{.}\) Consider the binary operation \(\oplus:\mathbb{Z}_{61}\times \mathbb{Z}_{61}\to \mathbb{Z}_{61}\) given by \(a \oplus b = \left(a + b\right) \bmod 61\text{.}\)
An element \(e\in \mathbb{Z}_{61}\) is the identity with respect to \(\oplus\) if \(a \oplus e= a\) and \(e \oplus a=a\) for all \(a\in \mathbb{Z}_{61}\text{.}\)
An element \(b\in \mathbb{Z}_{61}\) is the inverse of \(a\in \mathbb{Z}_{61}\) with respect to \(\oplus\) if \(a \oplus b=e\) and \(b \oplus a=e\text{.}\)
