A coset is a subset of a group.
Specifically, let be a group, and let be a subgroup of . The left cosets modulo are the subsets of of the form , for . Note that for any coset , the mapping is a bijection from to . Hence for any , .
The image of a left coset under the mapping is the right coset . This mapping induces a bijection from the set of left cosets of to the set of right cosets of .
The cardinality of the set of left cosets of is called the index of with respect to ; it is denoted . This is also the cardinality of the set of right cosets of .
Proposition. The relations , are equivalence relations.
Proof. We prove that the first relation is an equivalence relation; the second then follows by passing to the opposite law on .
We abbreviate as . For any , note that , so . If , then , so implies . Finally, if and , then ; hence and together imply . Hence is an equivalence relation.
Cosets and compatible relations
We call a relation left compatible with the group structure of if implies , for all . Similarly, we say is right compatible with the group structure of if implies . Note that is compatible with the group law on if and only if it is both left- and right-compatible with the structure.
Theorem. An equivalence relation on a group is left (resp. right) compatible with if and only if it is of the form (resp. ), for some subgroup of . In this case, is the equivalence class of , the identity, and the equivalence classes are the left (resp. right) cosets of .
Proof. We will consider only the case for left compatible with ; the other case follows from symmetry.
Let be the equivalence class of . Note that if and only if , which is true if and only if . It thus remains to show that is a subgroup of .
To this end, we note that evidently ; also, if , then , so . Finally, if are in , then . Thus is a subgroup of .
Conversely, suppose is a subgroup of , and define as . We have proven that is an equivalence relation; evidently if and only if . Now, if , then , so is left-compatible with the group structure of .
Now, if and only if ;. Hence the set of equivalent to (mod ) is the set . Thus the equivalence classes of are the left cosets mod .