Cosets

For G a group and H a subgroup, a right coset of H in G is

Similarly we can define a left coset

Typically, a left coset gH and a right coset Hg are not equal as sets. In additive notation, a right coset would be denoted as H+g and a left coset as g+H.

The right (left) cosets of a subgroup have the following properties

• if and only if .
• Either or else .
• Every element is an element of some right coset.

Similar properties hold for left cosets. We say that the cosets partition the group G. Indeed, they are the equivalence classes of the equivalence relation on G defined by (mod H) if and only if .

For example, the group consists of the 12 even permutations of . The set is a subgroup. The right cosets of H in are:

• H
• 
• 
• 

The left cosets of H in are:

• H
• 
• 
• 

Note that the number of elements in each coset is the same and that is not accidental. Also note that, other than the subgroup H, no left coset is a right coset.

The map defined by is a bijection. In particular for G a finite group, all cosets have the same number of elements. This is the basis for a proof of Lagrange's Theorem.

The number of cosets of H in G (right or left) is called the index and is denoted by . For G finite, we have . Thus the index multiplied by the order of the subgroup is the order of the group. Thus, for G finite, both the index and the order of the subgroup divide the order of G.

A right transversal for H in G is a set consisting of one element from each right coset. This is the same as a set of equivalence class representatives. A left transversal is defined similarly.

• Normal subgroups
• Factor groups