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.

The map gH→(gH)−1=Hg−1 defines a bijection between the left cosets and the right cosets of H, so the number of left cosets is equal to the number of right cosets. The common value is called the index of H in G.

For abelian groups, left cosets and right cosets are always the same. If the group operation is written additively then the notation used changes to g+H or H+g.

Cosets are a basic tool in the study of groups; for example they play a central role in Lagrange's theorem.