Normal subgroup

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Normal subgroup

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, more specifically in abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, a normal subgroup is a special kind of subgroup

Subgroup

In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *....
. Normal subgroups are important because they can be used to construct quotient group

Quotient group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
.

Évariste Galois

Évariste Galois

?variste Galois was a France mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a Necessary and sufficient conditions for apolynomial to be solvable by Nth root, thereby solving a long-standing problem....
was the first to realize the importance of the existence of normal subgroups.

Definitions
A subgroup, N, of a group, G, is called a normal subgroup if it is invariant under conjugation

Conjugation of isometries in Euclidean space

In a Group , the conjugate by g of h is ghg-1....
; that is, for each element, n, in N and each g in G, the element gng⁻¹ is still in N.

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Definitions

A subgroup, N, of a group, G, is called a normal subgroup if it is invariant under conjugation

Conjugation of isometries in Euclidean space

In a Group , the conjugate by g of h is ghg-1....
; that is, for each element, n, in N and each g in G, the element gng⁻¹ is still in N. We write

The following conditions are equivalent

Logical equivalence

In logic, statements p and q are logically equivalent if they have the same logical content.Syntax , p and q are equivalent if each can be proof from the other....
to requiring that a subgroup, N, be normal in G. Any one of them may be taken as the definition:

For all g in G, gNg⁻¹ ⊆ N.

For all g in G,gNg⁻¹ = N.*

The sets of left and right coset
Coset

In mathematics, if G is a group , H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G....
s of N in G coincide.

For all g in G, gN = Ng.*

N is a union
Union (set theory)

In set theory, the term Union refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets....
of conjugacy class
Conjugacy class

In mathematics, especially group theory, the elements of any group may be partition of a set into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure....
es of G.
There is some homomorphism
Group homomorphism

In mathematics, given two group and , a group homomorphism from to is a function h : G ? H such that for all u and v in G it holds that...
on G for which N is the kernel
Kernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective....
.

*These are logically stronger than the conditions above them and are not necessary for N to be a subgroup. They are properties of the subgroup.

Examples

and G are always normal subgroups of G. These groups are called the trivial subgroups, and if these are the only ones, then G is said to be simple
Simple group

In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself....
.

The center of a group
Center (group theory)

In abstract algebra, the center of a group G is the set Z of all elements in G which Commutative with all the elements of G. That is,...
is a normal subgroup.

The commutator subgroup
Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generating set of a group by all the commutators of the group....
is a normal subgroup.

More generally, any characteristic subgroup
Characteristic subgroup

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is closed under all automorphisms of the parent group ....
is normal, since conjugation is always an automorphism
Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of map the object to itself while preserving all of its structure....
.

All subgroups N of an abelian group
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
G are normal, because gN = Ng. A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group
Hamiltonian group

In group theory, a Dedekind group is a group G such that every subgroup of G is normal subgroup.All abelian groups are Dedekind groups....
.

The translation group in any dimension is a normal subgroup of the Euclidean group
Euclidean group

In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometry associated with the Euclidean Metric , are called Euclidean moves....
; for example in 3D rotating, translating, and rotating back results in only translation; also reflecting, translating, and reflecting again results in only translation (a translation seen in a mirror looks like a translation, with a reflected translation vector). The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances.

In the Rubik's Cube group
Rubik's Cube group

The Rubik's Cube provides a tangible representation of a group . The Rubik's Cube group can be thought of as the set of all cube operations with function composition as the group operation....
, the subgroup consisting of operations which only affect the corner pieces is normal, because no conjugate transformation can make such an operation affect an edge piece instead of a corner. By contrast, the subgroup consisting of turns of the top face only is not normal, because a conjugate transformation can move parts of the top face to the bottom and hence not all conjugates of elements of this subgroup are contained in the subgroup.

Properties

Normality is preserved upon surjective homomorphisms, and is also preserved upon taking inverse images.
Normality is preserved on taking direct products
A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation
Transitive relation

In mathematics, a binary relation R over a Set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
. However, a characteristic subgroup
Characteristic subgroup

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is closed under all automorphisms of the parent group ....
of a normal subgroup is normal. Also, a normal subgroup of a central factor is normal. In particular, a normal subgroup of a direct factor is normal.
Every subgroup of index 2 is normal. More generally, a subgroup H of finite index n in G contains a subgroup K normal in G and of index dividing n! called the normal core. In particular, if p is the smallest prime dividing the order of G, then every subgroup of index p is normal.

Lattice of normal subgroups

The normal subgroups of a group G form a lattice

Lattice (order)

In mathematics, a lattice is a partially ordered set in which subsets of any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain Axiom identity ....
under subset inclusion with least element and greatest element

Greatest element

In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S....
G. Given two normal subgroups N and M in G, meet is defined as and join is defined as

The lattice is complete

Complete lattice

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science....
and modular

Modular lattice

In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:Modular law: x = b implies x ? = ? b....
.

Normal subgroups and homomorphisms

Normal subgroups are of relevance because if N is normal, then the quotient group

Quotient group

In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element....
G/N may be formed: if N is normal, we can define a multiplication on cosets by

(a₂N) := (a₁a₂)N.

This turns the set of cosets into a group called the quotient group G/N. There is a natural homomorphism

Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ???? meaning "same" and ???f? meaning "shape"....
f : G → G/N given by f(a) = aN. The image f(N) consists only of the identity element of G/N, the coset eN = N.

In general, a group homomorphism f: G → H sends subgroups of G to subgroups of H. Also, the preimage of any subgroup of H is a subgroup of G. We call the preimage of the trivial group in H the kernel
Kernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective....
of the homomorphism and denote it by ker(f). As it turns out, the kernel is always normal and the image f(G) of G is always isomorphic to G/ker(f) (the first isomorphism theorem). In fact, this correspondence is a bijection between the set of all quotient groups G/N of G and the set of all homomorphic images of G (up to

Up to

In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e....
isomorphism). It is also easy to see that the kernel of the quotient map, f: G → G/N, is N itself, so we have shown that the normal subgroups are precisely the kernels of homomorphisms with domain

Domain (mathematics)

In mathematics, the domain of a given function is the set of "input" values for which the function is defined. For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 ....
G.

Normal subgroup

Definitions

Examples

Properties

Lattice of normal subgroups

Normal subgroups and homomorphisms

See also

External links