Question

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

Answer 1

Hint: 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 $*:Z \times Z \to Z$. The operation is defined as $m*n = m + n + 2$, where $m,n \in Z$.
We have to show that $\left( {Z,*} \right)$ is an abelian group.
We will check for all five properties which is to be satisfied to show that $\left( {Z,*} \right)$ is an abelian group.
Associative property:
Let $m,n,p \in Z$. Then $\left( {m*n} \right)*p = \left( {m + n + 2} \right)*p = \left( {m + n + 2 + p + 2} \right) = \left( {m + n + p + 4} \right)$
And, $m*\left( {n*p} \right) = m*\left( {n + p + 2} \right) = \left( {m + n + p + 2 + 2} \right) = \left( {m + n + p + 4} \right)$
Since $\left( {m*n} \right)*p = m*\left( {n*p} \right)$, we can say that associative property holds true.
Identity property:
Let $m \in Z$. Then we find the identity element $e \in Z$ as $m*e = m \Rightarrow m + e + 2 = m \Rightarrow e + 2 = 0 \Rightarrow e = - 2$.
Since, there exists an identity element for this operation, we can say that identity property holds true.
Inverse property:
Let $m \in Z$. Then we have to find the inverse of $m$, let us say $n \in Z$, such that $m*n = e$.
$m*n = e \Rightarrow m + n + 2 = - 2 \Rightarrow n = - 4 - m$
Since $m \in Z$, $\left( { - 4 - m} \right) \in Z$ and thus $n \in Z$.
Since the inverse of an element in $Z$ exists in $Z$, we can say that inverse property holds true.
Closure property:
Let $m,n \in Z$. We have to establish that $m*n \in Z$.
$m*n = m + n + 2$
Since $m$ and $n$ are integers, their sum will also be an integer and further sum with $2$ will also be an integer. Thus, $m + n + 2 \in Z$ and therefore $m*n \in Z$.
Since $m*n \in Z$, we can say that closure property holds true.
Commutative property:
Let $m,n \in Z$. Then we have to show that $m*n = n*m$.
$
  m*n = m + n + 2 \\
  n*m = n + m + 2 \\
 $
Since commutative property holds true for addition operation, we can write,
$n + m + 2 = m + n + 2$
Thus, $m*n = n*m$. Therefore commutative property also holds true.
Since all the five properties for an abelian group holds true for the operation $m*n = m + n + 2$ in $Z$, we can say that $\left( {Z,*} \right)$ 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.