Equivalence of definitions of left coset
This article gives a proof/explanation of the equivalence of multiple definitions for the term left coset
View a complete list of pages giving proofs of equivalence of definitions
The definitions that we have to prove as equivalent
We are given a group and a subgroup . We'd like to know whether a subset of is a left coset of . We want to show that the following three descriptions are equivalent:
- is in for any , and for any fixed , the map is a surjection from to
- There exists such that
- For any ,
We clearly have (3) implies (2) (because is nonempty). Let's show that (2) implies (1), and (1) implies (3) (the order of proof isn't really important, and once you see the proof, you'll see that it works all ways).
(2) implies (1)
Suppose . Then, pick elements . By definition and with . So the element is . This element is in .
Also, given any , we have and , so every occurs as for some choice of and . So the map from to is a surjection.
(1) implies (3)
Suppose it is true that for any . Then, pick any . We want to show that .
First, observe that . That's because given , , so .
We now want to show that . In other words, we want to show that any element of the form lives inside . But this follows from the fact that for any , there exists such that , so we get that .