Quotient group on arvau

Wikipedia / Quotient group

In the branch of mathematics known as group theory, a quotient group or factor group is a 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 and defining a group structure that operates on each such class (known as a congruence class) as a single entity.

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 is the normal subgroup. This is read as ' is short for modulo. (The notation should be interpreted with caution, as some authors (e.g., Vinberg for any is not normal in . Others (e.g., Dummit and Foote is normal in .)

Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image under a homomorphism is always isomorphic to a quotient of 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

Given a group and a subgroup , one can consider the corresponding left coset: . Cosets are a natural class of subsets of a group; for example consider the abelian group of integers, with operation of even integers. Then there are exactly two cosets: , which are the even integers, and .

For a general subgroup , it is desirable to define a compatible group operation on the set of all possible cosets, is normal if and only if holds for all is denoted .

Definition

be a normal subgroup of a group in , i.e. , with the group operation defined as . It can be shown that this operation satisfies all of the group axioms.

in . Since the identity element , .

, i.e. if and , then where . This proof is only valid for normal subgroups, as it uses the property .

is a normal subgroup. The definition however required be a normal subgroup. This is because the group operation is only well-defined for normal subgroups.

for any , then .

and . Using the property of cosets that , we have: .

, we have , and so is a normal subgroup.

Example: Addition modulo 6

For example, consider the group with addition modulo 6: . Consider the subgroup 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"

can be compared to division of integers. When dividing 12 by 3 one obtains the result 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, although one ends up with a group for a final answer instead of a number. In general groups have more structure than an arbitrary collection of objects: in the quotient 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 is abelian with addition modulo equals with addition modulo 2.

Example further explained...

when dividing by is odd.
By definition of , the kernel of , , is the set of all even integers.
Let is a subgroup, because the identity in , which is , is in are in contains its inverses.
is the quotient group of left cosets; .
Note that we have defined is even.
to .

Remainders of integer division

under addition. Let consisting of all multiples of is abelian. The cosets are the collection belongs to the coset by 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 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 is a set of the form are identical sets when the non-integer parts 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 (see Euler's identity).

Matrices of real numbers

real matrices real matrices with determinant (since it is the kernel of the determinant homomorphism is known as the special linear group .

Integer modular arithmetic

with addition modulo 4), and its subgroup is . This is a group with identity element , and group operations such as are isomorphic with .

Integer multiplication

Consider the multiplicative group of th residues is a multiplicative subgroup isomorphic to has the cosets . The Paillier cryptosystem is based on the conjecture without knowing the factorization of .

Properties

is isomorphic to the trivial group is isomorphic to .

The order of , by definition the number of elements, is equal to , the index in divided by the order of are infinite (for example, ).

There is a "natural" surjective group homomorphism belongs, that is: is sometimes called the onto . Its kernel is .

and the subgroups of containing is 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.

is abelian, nilpotent, solvable, cyclic or finitely generated, then so is .

is a subgroup in a finite group is one half of the order of exists and is isomorphic to is the smallest prime number dividing the order of a finite group, has order must be a normal subgroup of .

and a normal subgroup is a group extension by is a direct product or semidirect product 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 is also isomorphic to has only the trivial automorphism is different from and .

Quotients of Lie groups

is a Lie group is a normal and (topologically) closed Lie subgroup of is also a Lie group. In this case, the original group has the structure of a fiber bundle (specifically, a principal N-bundle and fiber equals .

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 of left cosets is not a group, but simply a differentiable manifold acts. The result is known as a homogeneous space.