Coset
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 
.
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