Find a subset of Z that is closed under addition but is not a group of the additive group Z.
Will the set of natural number be suffice?
Let S be the set of even natural numbers where S = {2, 4, 6, ...}. This will be closed with respect to addition because the sum of two even natural numbers is an even natural number.
However, since the set does not contain inverses and does not contain the identity element, it does not satisfy group properties.