DefinitionFor a subgroup of , and given some , we define
the first term is called a left coset and the second a right coset.
ExampleTake the symmetric group and subgroup then the cosets of are: and and
Cosets form partition elements. That is, for , the set of all left cosets is a partition of . Likewise for right-cosets.
All left- and right- cosets of have the same cardinality as . Since left- and right- cosets partition , then there are left cosets of and the same number of right cosets.
I will prove the first identity; the second is proven similarly. Assume that . Then exists some where . Multiplication gives . Since , then also . First note that if then . Now assume . Then , alternative proof. Assume . Then . Since and cosets are disjoint, then .