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Question: Let \(*\) be defined in \(Z\) by \(m*n = m + n + 2\). Show that \(\left( {Z,*} \right)\) is an abeli...

Let * be defined in ZZ by mn=m+n+2m*n = m + n + 2. Show that (Z,)\left( {Z,*} \right) is an abelian group?

Explanation

Solution

Groups are sets equipped with an operation satisfying certain basic properties. A group satisfies four properties – associativity, identity, inverse and closure property. An abelian group is a group in which the operation also satisfies the commutative property in the given set.

Complete step by step solution:
We have been given the definition of a binary operation :Z×ZZ*:Z \times Z \to Z. The operation is defined as mn=m+n+2m*n = m + n + 2, where m,nZm,n \in Z.
We have to show that (Z,)\left( {Z,*} \right) is an abelian group.
We will check for all five properties which is to be satisfied to show that (Z,)\left( {Z,*} \right) is an abelian group.
Associative property:
Let m,n,pZm,n,p \in Z. Then (mn)p=(m+n+2)p=(m+n+2+p+2)=(m+n+p+4)\left( {m*n} \right)*p = \left( {m + n + 2} \right)*p = \left( {m + n + 2 + p + 2} \right) = \left( {m + n + p + 4} \right)
And, m(np)=m(n+p+2)=(m+n+p+2+2)=(m+n+p+4)m*\left( {n*p} \right) = m*\left( {n + p + 2} \right) = \left( {m + n + p + 2 + 2} \right) = \left( {m + n + p + 4} \right)
Since (mn)p=m(np)\left( {m*n} \right)*p = m*\left( {n*p} \right), we can say that associative property holds true.
Identity property:
Let mZm \in Z. Then we find the identity element eZe \in Z as me=mm+e+2=me+2=0e=2m*e = m \Rightarrow m + e + 2 = m \Rightarrow e + 2 = 0 \Rightarrow e = - 2.
Since, there exists an identity element for this operation, we can say that identity property holds true.
Inverse property:
Let mZm \in Z. Then we have to find the inverse of mm, let us say nZn \in Z, such that mn=em*n = e.
mn=em+n+2=2n=4mm*n = e \Rightarrow m + n + 2 = - 2 \Rightarrow n = - 4 - m
Since mZm \in Z, (4m)Z\left( { - 4 - m} \right) \in Z and thus nZn \in Z.
Since the inverse of an element in ZZ exists in ZZ, we can say that inverse property holds true.
Closure property:
Let m,nZm,n \in Z. We have to establish that mnZm*n \in Z.
mn=m+n+2m*n = m + n + 2
Since mm and nn are integers, their sum will also be an integer and further sum with 22 will also be an integer. Thus, m+n+2Zm + n + 2 \in Z and therefore mnZm*n \in Z.
Since mnZm*n \in Z, we can say that closure property holds true.
Commutative property:
Let m,nZm,n \in Z. Then we have to show that mn=nmm*n = n*m.
mn=m+n+2 nm=n+m+2  m*n = m + n + 2 \\\ n*m = n + m + 2 \\\
Since commutative property holds true for addition operation, we can write,
n+m+2=m+n+2n + m + 2 = m + n + 2
Thus, mn=nmm*n = n*m. Therefore commutative property also holds true.
Since all the five properties for an abelian group holds true for the operation mn=m+n+2m*n = m + n + 2 in ZZ, we can say that (Z,)\left( {Z,*} \right) is an abelian group.

Note: To be an abelian group the given operation has to satisfy all the four properties of a group and an additional commutative property. All groups are not abelian groups. Only groups satisfying commutative property are abelian groups. The domain and range of the operation is important to check for the properties of the abelian group.