Show that set of integers Z is not a group under the binary operation multiplication. Justify each of the group properties that holds or not.

Show that set of integers Z is not a group under the binary operation multiplication.
Justify each of the group properties that holds or not.                                       

Solution:

        Here

                    The set of integers Z is not a group under the binary operation multiplication.

        Now, we have to justify each of the group properties that holds or not.

        Lets suppose a = 1, b = 2 and c = 3 where  `a, b \in Z` 

        So,

•  Commutative Property

            First, we discuss commutative property `\forall  `a,b \in Z`

                            a * b = 1 * 2 = 2 = 2 * 1 = b * a

            So, the commutative property is satisfied.

•  Associative Property

            Now, we discuss associative property `\forall a,b,c \in Z`

                            `a * (b*c) = 1 * (2*3)  = 6 = (1*2) * 3 = (a*b)*c`

            So, the associative property is satisfied.

•  Identity Property

            Now, we discuss identity property `\forall a \in Z`

                            `e*a = 1 * 1= 1 = 1 * 1 = a * e    where  \e = 1 \in Z`

            So, the identity property is satisfied.

•  Inverse Property

            Now, we discuss inverse property `\forall b \in Z`

                            `b = \frac{1}{b} \to 2 \ne\frac{1}{2}`                              

            So, the inverse property is not satisfied.

            So, its prove

                        The set of integers Z is not a group under the binary operation multiplication.