`  : Strictly speaking it is redundant, because, as you have said, a binary operation is defined on specific set S, and must be closed.  However, this is interesting to think about.  Strictly speaking, I think the operation ``+`` on Z and ``+`` on 2Z really are the same operation, since the elements in 2Z really are elements in Z.  It is not the case that Z contains an isomorphic copy of 2Z, where then the two ``+`` operations would really be different.  So then I think that if you have a subgroup, it actually isn't redundant to specify it must be closed under the given binary operation, since it really is the same operation as that of the larger group.  In any event, I think the little blurb about closure is wordy and I would not object to its removal.  `