Group structure on every nonempty set ($x\star y=j^{-1}(j(x)j(y))$)

Assuming choice, every nonempty set admits a group operation. A finite set receives a cyclic group structure. For an infinite set, a maximal partial group obtained by Zorn has infinite carrier $A$ with smaller complement; infinite-cardinal doubling implies the whole set is equipotent with $A$. Transport its operation along a bijection $j$. This final group need not extend a previously chosen partial group on its original subset; transport changes labels and only asserts existence.