Given any subset S of a group ⁠ G ⁠, the subgroup generated by S consists of all products of elements of S and their inverses. It is the smallest subgroup of G containing ⁠ S ⁠. In the example of symmetries of a square, the subgroup generated by r 2 and f v consists of these two elements, the identity element ⁠ i d ⁠, and the element ⁠ f h = f v ⋅ ⋅ r 2 ⁠. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.