Section 6.6
Cosets

Let G be a group and H a subgroup of G. For g G, we write

gH = {gh | h H}.


Consider the following relation ~ on G:

g ~ k if and only if k-1g H.

Lemma

The relation ~ is an equivalence relation. The ~-equivalence classes are the sets gH with g G.

The ~-equivalence classes of an equivalence relation partition G. These ~-equivalence classes are so important that they deserve a special name:

Definition

The ~-equivalence classes gH with g G, are called the left cosets of H in G.


The set of all the left cosets gH of H in G is denoted by G/H.