The left cosets of any subgroup H form a partition of ⁠ G ⁠ ; that is, the union of all left cosets is equal to G and two left cosets are either equal or have an empty intersection. The first case g 1 H = g 2 H happens precisely when ⁠ g 1 − − 1 ⋅ ⋅ g 2 ∈ ∈ H ⁠, i.e., when the two elements differ by an element of ⁠ H ⁠. Similar considerations apply to the right cosets of ⁠ H ⁠. The left cosets of H may or may not be the same as its right cosets. If they are (that is, if all g in G satisfy ⁠ g H = H g ⁠), then H is said to be a normal subgroup.