We define U(n) to be the set of all positive integers less than n and relatively prime to n. However, this definition does not state how to treat U(n) in terms of group theory.
Are we suppose to assume that any time we refer to U(n), it is to be treated as a group under multiplication modulo n? If not, where in our homework does it define how we should view U(n) in terms of groups?
Recall what Z//nZ is, namely the integers with usual multiplication and addition and subtraction with the understanding that n counts as zero.
Def: U(n) is the SUBSET of Z/nZ that consists of those cosets i+nZ where i is coprime to n. (NOTE: if i is coprime, then so is i+n and i+2n and ... so that this def makes sense). The group operation is the multiplication inherited from Z/nZ (and I really mean "multiplication", not "addition" here), which is in turn the multiplication inherited from Z.
If one is so inclined, one may view U(N) as indicated in the question: the set of all positive integers less than n and relatively prime to n. But hat is just a choice of coset representatives, nothing more.
So, as suggested in the question, "U(n)" really means "(U(n),*)" in all cases.
