Let be a group and . Define a relation on by if and only if . Prove that is an equivalence relation and describe for each the equivalence class . Use this to prove the following proposition.
Proposition: Let be a group and . Then (i) the set of left cosets of is a partition of and (ii) for all , if and only if .
- is an equivalence:
- Reflexive: , so that for all .
- Symmetric: Suppose . Then . Since is a subgroup, , and we have .
- Transitive: Suppose and . Then and . Thus we have and for some . Now , and substituting yields , hence . Thus .
So is an equivalence.
- Let and suppose . Then for some , hence . Thus ; so . Now suppose . Then for some , hence and we have , so that . Thus the -equivalence classes of are precisely the left cosets of .
- Proof of Proposition 4: The left cosets of form the equivalence classes of a relation on defined by if and only if ; the second conclusion of Proposition 4 follows trivially.