Quotient group

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 pronounced , where is short for modulo.)

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 illustrationEdit

Given a group   and a subgroup  , and an 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  .

DefinitionEdit

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 N is a normal subgroup. It still remains to be shown that this condition is not only sufficient but necessary to define the operation on G/N.

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,   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 6Edit

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:

 .

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"Edit

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.

ExamplesEdit

Even and odd integersEdit

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     / H  as   for   and   / H is the quotient group of left cosets;   / H .
Note that we have defined  ,   is   if   is odd and   if   is even.
Thus,   is an isomorphism from   / H to  .

Remainders of integer divisionEdit

A slight generalization of the last example. Once again consider the group of integers   under addition. Let n 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 1Edit

 
The cosets of the fourth roots of unity N in the twelfth roots of unity G.

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.

The real numbers modulo the integersEdit

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 numbersEdit

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 arithmeticEdit

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   are isomorphic with  .

Integer multiplicationEdit

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  .

PropertiesEdit

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  .[1]

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 groupsEdit

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 alsoEdit

NotesEdit

  1. ^ Dummit & Foote (2003, p. 120)
  2. ^ John M. Lee, Introduction to Smooth Manifolds, Second Edition, theorem 21.17

ReferencesEdit

  • Dummit, David S.; Foote, Richard M. (2003), Abstract Algebra (3rd ed.), New York: Wiley, ISBN 978-0-471-43334-7
  • Herstein, I. N. (1975), Topics in Algebra (2nd ed.), New York: Wiley, ISBN 0-471-02371-X