For a subgroup of a group and an element of , define to be the set and to be the set . A subset of of the form for some is said to be a left coset of and a subset of the form is said to be a right coset of .

For any subgroup , we can define an equivalence relation by if for some . The equivalence classes of this equivalence relation are exactly the left cosets of , and an element of is in the equivalence class . Thus the left cosets of form a partition of .

It is also true that any two left cosets of have the same cardinality, and in particular, every coset of has the same cardinality as , where is the identity element. Thus, the cardinality of any left coset of has cardinality the order of .

The same results are true of the right cosets of and, in fact, one can prove that the set of left cosets of has the same cardinality as the set of right cosets of .

This entry contributed by Nicolas Bray

CITE THIS AS:

Bray, Nicolas. "Coset." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Coset.html