A quotient group 
of 
by 
, where 
is a group and 
is a normal subgroup of
, is the set
of cosets of 
in 
. It is read "
modulo 
". Quotient groups are also called factor groups. The
elements of 
are written 
and form a group under the operation
induced by the group operation in 
. Thus,
 |
Since all elements of 
will appear in exactly one coset of the normal subgroup 
, it follows that
 |
where 
denotes the group order, interpreted as a cardinal
number for an infinite group. This is also a consequence
of Lagrange's group theorem with 
and 
, where 
is the identity element
of 
.
Although the slash notation conflicts with that for an extension field, the meaning can be determined based on context.
