## Cosets

If is a group, is a subgroup of and then is a left coset of in  is a right coset of in If or equivalently for all then is a normal subgroup of The left and right cosets coincide and the set of cosets forms a group with the group operation defined by For under the operation addition modulo 3, the cosets of are and If G is abelian then all subgroups are normal. if and only if is an element of H since as H is a subgroup, it must be closed and must contain the identity.

Any two left cosets of in are either identical or disjoint — i.e., the left cosets form a partition of such that every element of belongs to one and only one left coset. In particular the identity is in precisely one coset, and that coset is itself since this is also the only coset that is a subgroup.

The left cosets of in are the equivalence classes under the equivalence relation on given by if and only if and similarly for right cosets.

All left cosets and all right cosets have the same order (number of elements, or cardinality in the case of an infinite H), equal to the order of H (because H is itself a coset). 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 Lagrange's theorem allows us to compute the index in the case where and are finite, as per the formula:  