It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: 
the set of finite subsets of $S$ with the binary operation of *symmetric difference* forms a group, and in ZFC there is a bijection between $S$ and the set of finite subsets of $S$, so the group structure can be taken to $S$. However, the existence of this bijection needs the axiom of choice.

So my question is
> Can it be shown in ZF that for any non-empty set $S$ there exists a binary operation $*$ on $S$ making $(S,*)$ into a group?