H∩K is a nonempty set of G, and both H and K are closed for * operation, so it is also closed to treat the elements of H∩K as * operation. Both H and K are subgroups, which contain the identity of G and also the identity of H ∩ K. Any element in H ∩ K has an inverse in H and K, and Z is the inverse of the element in G respectively. Because of the uniqueness of the inverse, this inverse is in H ∩ K. In addition, the associative law holds. So H∩K is also a subgroup.