Definition Using Equivalence Classes
Some authors define the left cosets of H in G to be the equivalence classes under the equivalence relation on G given by x ~ y if and only if x−1y ∈ H. The relation can also be defined by x ~ y if and only if xh=y for some h in H. It can be shown that the relation given is, in fact, an equivalence relation and that the two definitions are equivalent. It follows that any two left cosets of H in G are either identical or disjoint. In other words every element of G belongs to one and only one left coset and so the left cosets form a partition of G. Corresponding statements are true for right cosets.
Read more about this topic: Coset
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