Quotient group explained
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of
and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as
group theory.
For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written, where
is the original group and
is the normal subgroup. This is read as '', where
is short for
modulo. (The notation should be interpreted with caution, as some authors (e.g., Vinberg
[1]) use it to represent the left cosets of
in
for
any subgroup
, even though these cosets do not form a group if
is not normal in
. Others (e.g., Dummit and Foote) only use this notation to refer to the quotient group, with the appearance of this notation implying the normality of
in
.)
Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of
. Specifically, the image of
under a homomorphism
is isomorphic to
where
denotes the kernel of .
The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects.
Definition and illustration
and a subgroup
, and a fixed element
, one can consider the corresponding left
coset: . Cosets are a natural class of subsets of a group; for example consider the
abelian group G of
integers, with
operation defined by the usual addition, and the subgroup
of even integers. Then there are exactly two cosets:, which are the even integers, and, which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation).
For a general subgroup, it is desirable to define a compatible group operation on the set of all possible cosets, . This is possible exactly when
is a normal subgroup, see below. A subgroup
of a group
is normal
if and only if the coset equality
holds for all . A normal subgroup of
is denoted .
Definition
Let
be a normal subgroup of a group . Define the set
to be the set of all left cosets of
in . That is, .
Since the identity element, . Define a binary operation on the set of cosets,, as follows. For each
and
in, the product of
and,, is . This works only because
does not depend on the choice of the representatives,
and, of each left coset,
and . To prove this, suppose
and
for some . Then
.
This depends on the fact that is a normal subgroup. It still remains to be shown that this condition is not only sufficient but necessary to define the operation on .
To show that it is necessary, consider that for a subgroup
of, we have been given that the operation is well defined. That is, for all
and for .
Let
and . Since, we have .
Now,
gN=(ng)N\LeftrightarrowN=(g-1ng)N\Leftrightarrowg-1ng\inN, \foralln\inN
and .
Hence
is a normal subgroup of .
It can also be checked that this operation on
is always associative,
has identity element, and the inverse of element
can always be represented by . Therefore, the set
together with the operation defined by
forms a group, the quotient group of
by .
Due to the normality of, the left cosets and right cosets of
in
are the same, and so,
could have been defined to be the set of right cosets of
in .
Example: Addition modulo 6
For example, consider the group with addition modulo 6: . Consider the subgroup, which is normal because
is
abelian. Then the set of (left) cosets is of size three:
G/N=\left\{a+N:a\inG\right\}=\left\{\left\{0,3\right\},\left\{1,4\right\},\left\{2,5\right\}\right\}=\left\{0+N,1+N,2+N\right\}
.
The binary operation defined above makes this set into a group, known as the quotient group, which in this case is isomorphic to the cyclic group of order 3.
Motivation for the name "quotient"
The reason
is called a quotient group comes from division of integers. When dividing 12 by 3 one obtains the answer 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, although we end up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects.
To elaborate, when looking at
with
a normal subgroup of, the group structure is used to form a natural "regrouping". These are the cosets of
in . Because we started with a group and normal subgroup, the final quotient contains more information than just the number of cosets (which is what regular division yields), but instead has a group structure itself.
Examples
Even and odd integers
Consider the group of integers
(under addition) and the subgroup
consisting of all even integers. This is a normal subgroup, because
is
abelian. There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group
is the cyclic group with two elements. This quotient group is isomorphic with the set
with addition modulo 2; informally, it is sometimes said that
equals the set
with addition modulo 2.
Example further explained...
Let
be the remainders of
when dividing by . Then,
when
is even and
when
is odd.
By definition of, the kernel of,, is the set of all even integers.
Let . Then,
is a subgroup, because the identity in, which is, is in, the sum of two even integers is even and hence if
and
are in,
is in
(closure) and if
is even,
is also even and so
contains its inverses.
Define
as
for
and
is the quotient group of left cosets; .
Note that we have defined,
is
if
is odd and
if
is even.
Thus,
is an isomorphism from
to .
Remainders of integer division
A slight generalization of the last example. Once again consider the group of integers
under addition. Let be any positive integer. We will consider the subgroup
of
consisting of all multiples of . Once again
is normal in
because
is abelian. The cosets are the collection . An integer
belongs to the coset, where
is the remainder when dividing
by . The quotient
can be thought of as the group of "remainders" modulo . This is a
cyclic group of order .
Complex integer roots of 1
The twelfth roots of unity, which are points on the complex unit circle, form a multiplicative abelian group, shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup
made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group into three cosets, shown in red, green and blue. One can check that the cosets form a group of three elements (the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.). Thus, the quotient group
is the group of three colors, which turns out to be the cyclic group with three elements.
Real numbers modulo the integers
Consider the group of real numbers
under addition, and the subgroup
of integers. Each coset of
in
is a set of the form, where
is a real number. Since
and
are identical sets when the non-
integer parts of
and
are equal, one may impose the restriction
without change of meaning. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. The quotient group
is isomorphic to the
circle group, the group of
complex numbers of
absolute value 1 under multiplication, or correspondingly, the group of
rotations in 2D about the origin, that is, the special
orthogonal group . An isomorphism is given by
(see
Euler's identity).
Matrices of real numbers
If
is the group of invertible
real
matrices, and
is the subgroup of
real matrices with
determinant 1, then
is normal in
(since it is the
kernel of the determinant
homomorphism). The cosets of
are the sets of matrices with a given determinant, and hence
is isomorphic to the multiplicative group of non-zero real numbers. The group
is known as the
special linear group .
Integer modular arithmetic
Consider the abelian group
(that is, the set
with addition
modulo 4), and its subgroup . The quotient group
is . This is a group with identity element, and group operations such as . Both the subgroup
and the quotient group
\left\{\left\{0,2\right\},\left\{1,3\right\}\right\}
are isomorphic with .
Integer multiplication
Consider the multiplicative group . The set
of th residues is a multiplicative subgroup isomorphic to . Then
is normal in
and the factor group
has the cosets . The
Paillier cryptosystem is based on the
conjecture that it is difficult to determine the coset of a random element of
without knowing the factorization of .
Properties
The quotient group
is
isomorphic to the
trivial group (the group with one element), and
is isomorphic to .
The order of, by definition the number of elements, is equal to, the index of
in . If
is finite, the index is also equal to the order of
divided by the order of . The set
may be finite, although both
and
are infinite (for example,).
There is a "natural" surjective group homomorphism, sending each element
of
to the coset of
to which
belongs, that is: . The mapping
is sometimes called the
canonical projection of
onto . Its
kernel is .
There is a bijective correspondence between the subgroups of
that contain
and the subgroups of ; if
is a subgroup of
containing, then the corresponding subgroup of
is . This correspondence holds for normal subgroups of
and
as well, and is formalized in the
lattice theorem.
Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.
If
is
abelian,
nilpotent,
solvable,
cyclic or
finitely generated, then so is .
If
is a subgroup in a finite group, and the order of
is one half of the order of, then
is guaranteed to be a normal subgroup, so
exists and is isomorphic to . This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if
is the smallest prime number dividing the order of a finite group,, then if
has order,
must be a normal subgroup of .
Given
and a normal subgroup, then
is a
group extension of
by . One could ask whether this extension is trivial or split; in other words, one could ask whether
is a
direct product or
semidirect product of
and . This is a special case of the extension problem. An example where the extension is not split is as follows: Let, and, which is isomorphic to . Then
is also isomorphic to . But
has only the trivial
automorphism, so the only semi-direct product of
and
is the direct product. Since
is different from, we conclude that
is not a semi-direct product of
and .
Quotients of Lie groups
If
is a
Lie group and
is a normal and closed (in the topological rather than the algebraic sense of the word) Lie subgroup of, the quotient
is also a Lie group. In this case, the original group
has the structure of a
fiber bundle (specifically, a
principal -bundle), with base space
and fiber . The dimension of
equals .
[2] Note that the condition that
is closed is necessary. Indeed, if
is not closed then the quotient space is not a
T1-space (since there is a coset in the quotient which cannot be separated from the identity by an open set), and thus not a
Hausdorff space.
For a non-normal Lie subgroup, the space
of left cosets is not a group, but simply a
differentiable manifold on which
acts. The result is known as a
homogeneous space.
See also
Notes and References
- Book: Vinberg, Ė B. . A course in algebra . 2003 . American Mathematical Society . 978-0-8218-3318-6 . Graduate studies in mathematics . Providence, R.I . 157.
- John M. Lee, Introduction to Smooth Manifolds, Second Edition, theorem 21.17