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).