Cosets are either identical or disjoint

Let be a group, , and be a subgroup. #m/thm/group Then the left cosets are

identical iff
disjoint otherwise

Proof

Let , and be a subgroup. Due to basic Properties,

Next assume there exist such that , i.e. and have a common element. Then , whence and since , it follows and thus . Hence is , and can share no common element.

#state/tidy | #lang/en | #SemBr