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Given a non-empty set $X$. How do we plug a binary operation $*$ on the set $X$ so that $(X,*)$ forms a group?
There are sets, namely the set of natural numbers, set of prime numbers, set of perfect numbers and many for which finding an * to form them group is an extremely difficult job. Although I know some of the * for which the set of natural number forms a Group with repsect to *.
Please correct me if my understanding goes wrong!
Suppose that $X$ and $Y$ are nonempty sets of equal cardinality. If $(Y, \circ)$ is a group, then we can use a bijection $f: X \rightarrow Y$ to construct a binary operation $*$ on $X$ so that $(X, *)$ is a group and $(X, *) \cong (Y, \circ)$. Proving this is a good exercise.
Thus if $X$ is finite, the fact that $X$ has the same cardinality as the cyclic group of order $|X|$ gives us a binary operation $*$ that makes $(X, *)$ into a cyclic group.
What if $X$ is infinite? In this case we need the Axiom of Choice (see this question from MO). It is a consequence of choice that $X$ has the same cardinality as the set $\mathscr{R}$ of finite subsets of $X$. Now the set $\mathscr{R}$ equipped with symmetric difference makes $\mathscr{R}$ into an abelian group, so we are done.
In fact, $\mathscr{R}$ has a nontrivial structure of a ring without unit, when we define the sum of two sets as their symmetric difference, and the product of two sets as their intersection. Note also that finite cyclic groups have a ring structure (with unit).
It depends on the type of group you want. In the case of a collection of integers/rationals, one could create groups akin to $\mathbb{Z}_n$ or $\mathbb{Z}/\langle n \rangle$ or $(\mathbb{Z},+)$ or maybe even $(\mathbb{Q},\cdot)$. However, with many collections of numbers, it is not obvious how to make a group out of them. There are the 'obvious' ways, as I just mentioned. There are also very suprising ways to form groups: the easy example is the way we form a group out of the rational points on elliptic curves (assuming there is a rational point).
For your example concerning a group over the primes, you may want to read a question I asked previously, Are there Groups of Strictly Primes. In general, however, there is no right/wrong way to form a group out of a collection of numbers. In fact, the Axiom of Choice allows us to form a group out of any nonempty collection of numbers of any cardinality. So we know there is always at least one group given a nonempty collection of elements.
