Show that if $H$ is a subgroup of the group $(G, *)$ that contains all natural numbers $k\ge4$ then $H$ contains all rational numbers $q>3$.

Question

Consider the group $(G, *)$ with underlying set $G = (3, \infty)\subset\Bbb R\,$ and operation

$$x * y = (x-3)(y-3)+3$$

I have to show that if $H$ is a subgroup of the group $(G, *)$ that contains all natural numbers $k \ge 4$, then $H$ contains all rational numbers $q > 3$.

I'm not really sure how to interpret this. What is asked, really? And how to solve that? I mean, if $H$ contains only natural numbers that are greater that $4$, wouldn't the binary operation:

$$x * y = (x - 3)(y - 3) + 3$$

yield natural numbers only, since $x$ and $y$ are natural numbers and greater than $4$? Where do rational numbers come into the picture? And how can I show what is asked?

Answer 1

Compute the identity element $e$ and furthermore the inverse $x^{-1}$ of an element $x$. That should give you an argument of why the group contains rational numbers.