General Properties
The identity is in precisely one left or right coset, namely H itself. Thus H is both a left and right coset of itself.
A coset representative is a representative in the equivalence class sense. A set of representatives of all the cosets is called a transversal. There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here. Some books on very applied group theory erroneously identify the conjugacy class as 'the' equivalence class as opposed to a particular type of equivalence class.
Read more about this topic: Coset
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