Forgetful functor from groups to pointed sets is surjective

Verbal statement
Given any set with a chosen point, we can give a group structure to the set such that the chosen point is the identity element.

Equivalently, every nonzero cardinal arises as the order of some group, or, every nonempty set admits a group structure.

Category-theoretic statement
The forgetful functor from the category of groups to the category of pointed sets, that sends a group to its underlying set with the identity element as the chosen point, is surjective.

The finite case
Every finite cardinal, i.e., every positive integer, occurs as the order of a cyclic group. Specifically, the integer $$n$$ occurs as the order of the cyclic group of order $$n$$, i.e., the group $$\mathbb{Z}/n\mathbb{Z}$$, or $$C_n$$ -- the additive group of integers modulo $$n$$.

The infinite case
For the infinite case, we use the following fact: Every infinite cardinal equals its product with any finite or countable cardinal.

Thus, for an infinite cardinal $$\kappa$$, we can prove that $$\kappa$$ equals the cardinality of a restricted direct product of $$\kappa$$ copies of a countable group. In particular, it equals the cardinality of a vector space over a countable field with basis of size $$\kappa$$, as also the cardinality of the free Abelian group on $$\kappa$$ generators.