Question

Show that $\left( Z,* \right)$ is an infinite abelian group, where ‘*’ is defined as:
$a*b=a+b+2$ , and $Z$ is the set of all integers.

Answer 1

Hint: First check all the four fundamental properties of a group. Which are closure property, associativity, the identity property, and the inverse property. Then check if the operation is commutative or not.

Complete step-by-step answer:

We can say that $\left( Z,* \right)$ is a group if it satisfies the following four properties:
Closure property: If we take any two elements from the set, which is $Z$ here, then the composition of those two elements also should belong from that set. That is, $a,b\in Z\Rightarrow a*b\in Z$.
Associativity: The defined operation should be associative. That is, $\left( a*b \right)*c=a*\left( b*c \right)$.
The identity property: There exist an element, say $e$ , such that: $a*e=e*a$ , for all $a\in Z$.
The inverse property: For each $a\in Z$, there exist an element ${{a}^{-1}}$ in $Z$ such that: $a*{{a}^{-1}}={{a}^{-1}}*a=e$
$\left( Z,* \right)$ is said to be an abelian group if for all $a,b\in Z$, $a*b=b*a$.
Now let us check the first four properties one by one.
Let, $a,b\in Z$.
$a*b=a+b+2$ also belongs to the set of integers. Because if a and b are integers then addition of this two integers will give us an integer and if we add 2 with them it will again give us an integer. Therefore the closure property holds.
Let $a,b,c\in Z$
$\begin{aligned} & a*b=a+b+2 \\\ & \left( a*b \right)*c=\left( a*b \right)+c+2=a+b+2+c+2=a+b+c+4 \\\ \end{aligned}$
$\begin{aligned} & b*c=b+c+2 \\\ & a*\left( b*c \right)=a+\left( b*c \right)+2=a+b+c+2+2=a+b+c+4 \\\ \end{aligned}$
Therefore:
$a*\left( b*c \right)=\left( a*b \right)*c$, so the associativity holds.
Let us assume that for every element that belongs to the set, there exist an element e such that:
$\begin{aligned} & a*e=a \\\ & \Rightarrow a+e+2=a \\\ & \Rightarrow e+2=a-a \\\ & \Rightarrow e+2=0 \\\ & \Rightarrow e=-2 \\\ \end{aligned}$
So the identity element exists and that is -2.
Let us assume that for each element $a\in Z$ there exist an element ${{a}^{-1}}$ such that:
$\begin{aligned} & a*{{a}^{-1}}=e \\\ & \Rightarrow a+{{a}^{-1}}+2=-2 \\\ & \Rightarrow a+{{a}^{-1}}=-2-2 \\\ & \Rightarrow {{a}^{-1}}=-4-a \\\ \end{aligned}$
So for each element the inverse element exists.
Therefore $\left( Z,* \right)$ satisfies all the four properties. Hence $\left( Z,* \right)$ is a group.
Now we have to check if it is abelian or not.
$a*b=a+b+2=b+a+2=b*a$
Therefore for all $a,b\in Z$
$a*b=b*a$
Hence $\left( Z,* \right)$ is an abelian group.
We know that there are infinite numbers of elements in the set of all integers.
Hence, $\left( Z,* \right)$ is an infinite abelian group.

Note: We generally make mistakes while checking the four properties. To be a group all the four properties have to be satisfied. Then we need to check the abelian property separately. A group is not always abelian.