Cosets Partition the Group
If , the left cosets of partition : they cover , and any two left cosets that intersect are equal.
Lemma 3.3
If , the left cosets partition . That is,
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If , then for any
Proof
For any , we have since , so the left cosets cover . Conversely, every element of a left coset lies in .
If , choose with . Then , so . For any , , hence . Symmetry gives equality.
Related
Stated in
- Lemma 3.3§3.1 Cosets