prove that set of all non zero rational numbers is a group with respect to product

Question : Prove that set of all non zero rational numbers is a group with respect to product.

Proof :

We shall prove that set of all non zero rational numbers Q\{0} satisfies all the axioms of a group.

1. Closure axiom

Product of two rational numbers is a again a rational number

for example (1/2).(2/3)=1/3 , (-1/3).(4)= -4/3, (-1/2).(-3/2) =3/4

• Associative axiom

Product of two rational numbers is associative

for example -1.((1/2).3)= -3/2 and  ((-1).(1/2)).3 =-3/2=6 that is -1.((1/2).3)= ((-1).(1/2)).3

• Existence of identity 

(½).1 =1/2 , 2.(1)=2, – 4.(1) =- 4

So 1 is multiplicative identity.

• Existence of inverse

1/4  . (4) = 1, 2. (1/2) =1 , -6.(-1/6)  = 1

Thus multiplicative inverse of a non zero rational number is reverse of it.

Thus (Q\{0}, .) is a group.