Equivalence of definitions of coset

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


 * 1) It occurs as the fact about::left coset of a subgroup.
 * 2) It occurs as the fact about::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.