For N a normal subgroup of G we can define an associative binary operation on the set of cosets of N in G which is inherited from the binary operation on G.
Let G/N denote the set of cosets of N in G.
The elements in G/N are of the form Ng where 
. Of course we
could name the element Ng as Nx, say, provided Nx = Ng or that
. This is often a confusing point - the name of a coset is
not unique - but that is often useful.
Now define the binary operation on G/N by 
.
This definition is independent of the name we used for the coset. Moreover,
this is an associative binary operation and relative to it, there is an
identity and for each element, an inverse. The identity is N. The
inverse of Ng is 
.
The set G/N under this binary operation is a group known as a
factor group of G. The size of G/N is of course given by
. For a finite group G, the number of elements in
G/N is 
.
As an example, let's take 
where r is a rotation of 
, x reflection about the x-axis,
y reflection about the y-axis, d reflection about the diagonal
y=x and a reflection about the diagonal y=-x. The subgroup
is a normal subgroup of 
. The cosets are
N, 
, 
and 
.
Just as easily, we could have named the cosets 
, 
, Ny and Na.
In fact there are 16 ways to name the cosets but we'll work with
. Then in 
, (Nx)(Nr) = Nxr. However,
Nxr is not one of the names we have chosen for a coset. So from the
multiplication table in we see that Nxr = Nd since xr=d. This is
still not one of the names we have chosen. Finally, since a and d are in
the same coset we have NxNr = Nxr = Nd = Na. Continue in this way to get
the following multiplication table for .
On a point of notation, it is often easier to denote Ng by 
and N by 1.
This still distinguishes the element of the factor group from the element of
the group but is a lot easier to write than Ng. But to abuse notation
even more, once we know we are in the factor group, it is often convenient to
drop the bar and denote the coset merely by g. This is not recommended
until you have more experience. However, having said that, our table above
would be written as follows.
As another example, the set 
is a
normal subgroup of 
under addition. We shall restrict our
discussion to the case n=4.
The cosets are 
, 
, 
and 
.
The inherited binary operation is now denoted by + and the addition table
for 
is
You might recognize this table as addition modulo 4. In the case of
the factor group has elements through 
and the binary operation is addition modulo n.