Step 1:

Let $G=Q^+$ the set of all positive rationals.Let $*$ is the operation defined by $a*b=\large\frac{ab}{3}$ for $a,b\in Q^+$

Closure : Let $a,b\in Q^+$

$a*b=\large\frac{ab}{3} $$\in Q^+$

Since the product of two positive rational numbers is again a positive rational number.

The closure property is satisfied.

Step 2:

Associativity :

$(a*b)*c= (\large\frac{ab}{3})*c$

$\qquad\qquad= \large\frac{abc}{9}$

$\qquad\qquad=a* \large\frac{bc}{3}$

$\qquad\qquad= a*(b*c)$

The associative property is satisfied.

Step 3:

Existence of identity :

Consider $a,e\in Q^+$ such that

$a*e=a$

$\large\frac{ae}{3}$$=a$

$e=3$

Now $a*3=3*a=a$ for $a\in Q^+$

$\therefore 3$ is the identity element.

Step 4:

Existence of inverse :

Let $a\in Q^+$

Consider $a'$ such that $a*a'=3$

$\Rightarrow \large\frac{aa'}{3}$$=3\Rightarrow a'=\large\frac{9}{a}$$\in Q^+(a\neq 0)$

It can be seen that for $a'=\large\frac{9}{a}$$,a'a=aa'=3$

$\therefore$ every element in $Q^+$ has an inverse.

The four group axioms being satisfied,$(Q^+,*)$ is a group.