Given a group and a subgroup , the left cosets of in partition , in the sense that every element of is in precisely one coset.
Firstly, every element is in a coset: since for any . So we must show that no element is in more than one coset.
Suppose is in both and . Then we claim that , so in fact the two cosets and were the same. Indeed, , so there is such that . Therefore .
Exercise: .
Suppose is in the left-hand side. Then it is in the right-hand side immediately: letting .
Conversely, suppose is in the right-hand side. Then we may write , so is in the right-hand side; but then is in so this is exactly an object which lies in the left-hand side.
But that is just .
By repeating the reasoning with and interchanged, we have ; this completes the proof.
The fact that the left cosets partition the group means that we can, in some sense, "compress" the group with respect to . If we are only interested in "up to" , we can deal with the partition rather than the individual elements, throwing away the information we're not interested in.
This concept is most importantly used in defining the quotient group. To do this, the subgroup must be normal (proof). In this case, the collection of cosets itself inherits a group structure from the parent group , and the structure of the quotient group can often tell us a lot about the parent group.