Forgetful functor from groups to pointed sets is surjective

Statement

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.

Proof

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.