Group theory MOC

Coset

Given a subgroup , the left coset of in an element is defined as group

likewise the right coset as

Properties

1. iff ( by Reärrangement lemma, by )
2. As a consequence of the Reärrangement lemma .
3. Cosets are either identical or disjoint.
4. Every is contained in at least one coset of , namely (by )
5. From 2–4, may be partitioned into equally sized cosets. Hence the order of a subgroup divides the order of a group.

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