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.