Normal subgroup

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In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup $N$ of the group $G$ is normal in $G$ if and only if $gng^ \in N$ for all $g \in G$ and $n \in N.$ The usual notation for this relation is $N \triangleleft G.$ Normal subgroups are important because they (and only they) can be used to construct
quotient group A quotient group or factor group is a mathematical group (mathematics), 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 "factor ...
s of the given group. Furthermore, the normal subgroups of $G$ are precisely the kernels of group homomorphisms with domain $G,$ which means that they can be used to internally classify those homomorphisms.
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, ...
was the first to realize the importance of the existence of normal subgroups.

# Definitions

A
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
$N$ of a group $G$ is called a normal subgroup of $G$ if it is invariant under conjugation; that is, the conjugation of an element of $N$ by an element of $G$ is always in $N.$ The usual notation for this relation is $N \triangleleft G.$

## Equivalent conditions

For any subgroup $N$ of $G,$ the following conditions are equivalent to $N$ being a normal subgroup of $G.$ Therefore, any one of them may be taken as the definition: * The image of conjugation of $N$ by any element of $G$ is a subset of $N.$ * The image of conjugation of $N$ by any element of $G$ is equal to $N.$ * For all $g \in G,$ the left and right cosets $gN$ and $Ng$ are equal. * The sets of left and right cosets of $N$ in $G$ coincide. * The product of an element of the left coset of $N$ with respect to $g$ and an element of the left coset of $N$ with respect to $h$ is an element of the left coset of $N$ with respect to $g h$: for all $x, y, g, h \in G,$ if $x \in g N$and $y \in h N$ then $x y \in \left(g h\right) N.$ * $N$ is a union of
conjugacy class In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
es of $G.$ * $N$ is preserved by the
inner automorphism In abstract algebra an inner automorphism is an automorphism of a Group (mathematics), group, Ring (mathematics), ring, or Algebra over a field, algebra given by the Conjugacy class#Conjugacy as group action, conjugation action of a fixed element, ...
s of $G.$ * There is some
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
$G \to H$ whose
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
is $N.$ * There is some
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the ...
on $G$ for which the
equivalence class In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
of the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a Set (mathematics), set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in alge ...
is $N$. * For all $n\in N$ and $g\in G,$ the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
is in $N.$ * Any two elements commute regarding the normal subgroup membership relation. That is, for all $g, h \in G,$ $g h \in N$ if and only if $h g \in N.$

# Examples

For any group $G,$ the trivial subgroup $\$ consisting of just the identity element of $G$ is always a normal subgroup of $G.$ Likewise, $G$ itself is always a normal subgroup of $G.$ (If these are the only normal subgroups, then $G$ is said to be simple.) Other named normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup. If $G$ is an
abelian group In mathematics, an abelian group, also called a commutative group, is a group (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are w ...
then every subgroup $N$ of $G$ is normal, because $gN = \_ = \_ = Ng.$ A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group. A concrete example of a normal subgroup is the subgroup $N = \$ of the
symmetric group In abstract algebra, the symmetric group defined over any set (mathematics), set is the group (mathematics), group whose Element (mathematics), elements are all the bijections from the set to itself, and whose group operation is the function c ...
$S_3,$ consisting of the identity and both three-cycles. In particular, one can check that every coset of $N$ is either equal to $N$ itself or is equal to $\left(12\right)N = \.$ On the other hand, the subgroup $H = \$ is not normal in $S_3$ since $\left(123\right)H = \ \neq \ = H\left(123\right).$ This illustrates the general fact that any subgroup $H \leq G$ of index two is normal. In the Rubik's Cube group, the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal. The translation group is a normal subgroup of the
Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometry, isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transfo ...
in any dimension. This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation. By contrast, the subgroup of all
rotations Rotation, or spin, is the circular movement of an object around a ''axis of rotation, central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A t ...
about the origin is ''not'' a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin.

# Properties

* If $H$ is a normal subgroup of $G,$ and $K$ is a subgroup of $G$ containing $H,$ then $H$ is a normal subgroup of $K.$ * 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
. The smallest group exhibiting this phenomenon is the
dihedral group In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of order 8. However, a characteristic subgroup of a normal subgroup is normal. A group in which normality is transitive is called a T-group. * The two groups $G$ and $H$ are normal subgroups of their
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying Set (mathematics), sets, together with a suitably defined structure on the product set. More ...
$G \times H.$ * If the group $G$ is a
semidirect product In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
$G = N \rtimes H,$ then $N$ is normal in $G,$ though $H$ need not be normal in $G.$ * If $M$ and $N$ are normal subgroups of an additive group $G$ such that $G = M + N$ and $M \cap N = \$, then $G = M \oplus N.$ * Normality is preserved under surjective homomorphisms; that is, if $G \to H$ is a surjective group homomorphism and $N$ is normal in $G,$ then the image $f\left(N\right)$ is normal in $H.$ * Normality is preserved by taking inverse images; that is, if $G \to H$ is a group homomorphism and $N$ is normal in $H,$ then the inverse image $f^\left(N\right)$ is normal in $G.$ * Normality is preserved on taking direct products; that is, if $N_1 \triangleleft G_1$ and $N_2 \triangleleft G_2,$ then $N_1 \times N_2\; \triangleleft \;G_1 \times G_2.$ * Every subgroup of
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
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. * The fact that normal subgroups of $G$ are precisely the kernels of group homomorphisms defined on $G$ accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
, and a group is
imperfect The imperfect (list of glossing abbreviations, abbreviated ) is a verb form that combines past tense (reference to a past time) and imperfective aspect (reference to a continuing or repeated event or state). It can have meanings similar to the Eng ...
if and only if the derived subgroup is not supplemented by any proper normal subgroup.

## Lattice of normal subgroups

Given two normal subgroups, $N$ and $M,$ of $G,$ their intersection $N\cap M$and their product $N M = \$ are also normal subgroups of $G.$ The normal subgroups of $G$ form a lattice under subset inclusion with
least element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually, ...
, $\,$ and greatest element, $G.$ The meet of two normal subgroups, $N$ and $M,$ in this lattice is their intersection and the join is their product. The lattice is complete and
modular Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
.

# Normal subgroups, quotient groups and homomorphisms

If $N$ is a normal subgroup, we can define a multiplication on cosets as follows: $\left(a_1 N\right) \left(a_2 N\right) := \left(a_1 a_2\right) N.$ This relation defines a mapping $G/N\times G/N \to G/N.$ To show that this mapping is well-defined, one needs to prove that the choice of representative elements $a_1, a_2$ does not affect the result. To this end, consider some other representative elements $a_1\text{'}\in a_1 N, a_2\text{'} \in a_2 N.$ Then there are $n_1, n_2\in N$ such that $a_1\text{'} = a_1 n_1, a_2\text{'} = a_2 n_2.$ It follows that $a_1' a_2' N = a_1 n_1 a_2 n_2 N =a_1 a_2 n_1' n_2 N=a_1 a_2 N,$where we also used the fact that $N$ is a subgroup, and therefore there is $n_1\text{'}\in N$ such that $n_1 a_2 = a_2 n_1\text{'}.$ This proves that this product is a well-defined mapping between cosets. With this operation, the set of cosets is itself a group, called the
quotient group A quotient group or factor group is a mathematical group (mathematics), 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 "factor ...
and denoted with $G/N.$ There is a natural
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

, $f : G \to G/N,$ given by $f\left(a\right) = a N.$ This homomorphism maps $N$ into the identity element of $G/N,$ which is the coset $e N = N,$ that is, $\ker\left(f\right) = N.$ In general, a group homomorphism, $f : G \to H$ 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 may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of the homomorphism and denote it by $\ker f.$ As it turns out, the kernel is always normal and the image of $G, f\left(G\right),$ is always
isomorphic In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

to $G / \ker f$ (the
first isomorphism theorem In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
). In fact, this correspondence is a bijection between the set of all quotient groups of $G, G / N,$ and the set of all homomorphic images of $G$ (
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' wi ...
isomorphism). It is also easy to see that the kernel of the quotient map, $f : G \to G/N,$ is $N$ itself, so the normal subgroups are precisely the kernels of homomorphisms with domain $G.$

# Normal subgroups and Sylow Theorem

The Second Sylow Theorem states: If $P$ and $K$ are Sylow p-subgroups of a group $G$, then there exists $x \in G$ such that $P = x^Kx.$ There is a direct corollary of the theorem above: Let $G$ be a finite group and $K$ a Sylow p-subgroup for some prime $p$. Then $K$ is normal in $G$ if and only if $K$ is the only Sylow p-subgroup in $G$.

## Operations taking subgroups to subgroups

*
Normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group (mathematics), group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutativity, com ...
* Conjugate closure * Normal core

## Subgroup properties complementary (or opposite) to normality

* Malnormal subgroup * Contranormal subgroup * Abnormal subgroup * Self-normalizing subgroup

## Subgroup properties stronger than normality

* Characteristic subgroup * Fully characteristic subgroup

## Subgroup properties weaker than normality

* Subnormal subgroup * Ascendant subgroup * Descendant subgroup * Quasinormal subgroup * Seminormal subgroup * Conjugate permutable subgroup * Modular subgroup * Pronormal subgroup * Paranormal subgroup * Polynormal subgroup * C-normal subgroup

## Related notions in algebra

*
Ideal (ring theory) In ring theory, a branch of abstract algebra, an ideal of a ring (mathematics), ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction ...

# References

* * * * * * * * * * * *

* I. N. Herstein, ''Topics in algebra.'' Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.