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 
.
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 cardinal number
as the set of right cosets of 
.
