Remark

Just as we have defined left cosets we can also define right cosets of a subgroup H in a group G. These are the sets

Hg = {hg | h H}.

Unfortunately, the notation for the set of right cosets is somewhat confusing; it is denoted by H\G, which is the same notation as the complement of H in G. The notation H\\G for the cosets is sometimes used to distinguish the two.

Right cosets are the equivalence classes of the relation ~^r defined by

g ~^r h if and only if hg^-1 H.

If G is not commutative, they need not coincide with left cosets!