There exist groups of every order

Statement
For every nonzero cardinal $$\kappa$$ (finite or infinite), there exists a group whose order equals the cardinal $$\kappa$$.

Note that for finite cardinals, i.e., finite numbers, we can take the cyclic group, so the statement is interesting for infinite cardinals.

The proof of the statement assumes, and is equivalent to, the axiom of choice in the Zermelo-Fraenkel model of set theory.

Online discussions

 * Math Overflow discussion of the existence of groups of every order in models of ZF