Equivalence of definitions of coset

This article gives a proof/explanation of the equivalence of multiple definitions for the term coset
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a nonempty subset of a group:

1. It occurs as the Left coset of a subgroup (?).
2. It occurs as the Right coset of a subgroup (?).
3. Its left translates form a partition of the group.
4. Its right translates form a partition of the group.