Coset

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.

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