Coset

A coset is a left or right coset of some subgroup in G. Since Hg = g ( g−1Hg ), the right cosets Hg (of H ) and the left cosets g ( g−1Hg ) (of the conjugate subgroup g−1Hg ) are the same. Hence it is not meaningful to speak of a coset as being left or right unless one first specifies the underlying subgroup. In other words: a right coset of one subgroup equals a left coset of a different (conjugate) subgroup. If the left cosets and right cosets are the same then H is a normal subgroup and the cosets form a group called the quotient group.