We have gH = H if and only if g is an element of H. Any two left cosets are either identical or disjoint. The left cosets form a partition of G: every element of G belongs to one and only one left coset. The left cosets of H in G are the equivalence classes under the equivalence relation on G given by x ~ y if and only if x -1y ∈ H. All these statements are also true for right cosets.
All left cosets and all right cosets have the same number of elements (or cardinality in the case of an infinite H). Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of H in G, written as [G : H]. The following useful formula often allows to compute the index:
Therefore, if G is a finite group, then the number of left cosets of H can be calculated by dividing the order of G by the order of H. The same is true for the number of right cosets of H.