Based on this answer on Quora, group theory doesn't need set theory to be formulated. But isn't it hardwired into the definition of a group that a group is an ordered pair $(G,\; \cdot\;)$, where G is a set. Furthermore, to even define binary operations, would we not need the notion of sets, since functions are a map from a set to another set.
Is it because the term "set" used in the definition of a group is just in the sense of "a collection of objects", rather than a set in ZFC.
