# Coset

In

mathematics, if

*G* is a

group,

*H* a

subgroup of

*G* and

*g* an element of

*G*, then

*gH* = { *gh* : *h* an element of *H* } is a **left coset** of *H* in *G*

and

*Hg* = { *hg* : *h* an element of *H* } is a **right coset** of *H* in *G*.

### Some properties

We have *gH* = *H* if and only if *g* is an element of *H*.
Any two left cosets are either identical or disjoint. The left cosets form a partition of *G*: every element of *G* belongs to one and only one left coset.
The left cosets of *H* in *G* are the equivalence classes under the equivalence relation on *G* given by *x* ~ *y* if and only if *x*^{ -1}*y* ∈ *H*. All these statements are also true for right cosets.

All left cosets and all right cosets have the same number of elements (or cardinality in the case of an infinite *H*). 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 [*G* : *H*]. The following useful formula often allows to compute the index:

- |
*G*| = [*G* : *H*] · |*H*|

(where |

*G*| and |

*H*| denote the cardinality of the two groups). See

theorem of Lagrange for a proof.

Therefore, if *G* is a finite group, then the number of left cosets of *H* can be calculated by dividing the order of *G* by the order of *H*. The same is true for the number of right cosets of *H*.

The subgroup *H* is normal if and only if the left coset *gH* is equal to the right coset *Hg*, for all *g* in *G*. In this case one can turn the set of all cosets into a group, the factor group of *G* by *H*.