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 cardinal number, and in particular, every coset of has the same cardinal number as , where is the identity element. Thus, the cardinal number of any left coset of has cardinal number the order of .