TheInfoList

OR:

In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, 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 ''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, ''H'' is a subgro ...
that is invariant under
conjugation Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
by members of the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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 groups of the given group. Furthermore, the normal subgroups of $G$ are precisely the kernels of group homomorphisms with
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
$G,$ 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 radicals, ...
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 ''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, ''H'' is a subgro ...
$N$ of a group $G$ is called a normal subgroup of $G$ if it is invariant under
conjugation Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
; 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 Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *'' Equiv ...
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 classes of $G.$ * $N$ is preserved by the
inner automorphism In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
s of $G.$ * There is some group homomorphism $G \to H$ whose kernel is $N.$ * There is some congruence relation on $G$ for which the equivalence class of the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a 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 algebraic structures ...
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 Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
.) 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 in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
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 is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
$S_3,$ 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 In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every ...
is a normal subgroup of the
Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) 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 transformations ...
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 '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A ho ...
. The smallest group exhibiting this phenomenon is the
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
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 $G \times H.$ * If the group $G$ is a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way 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 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 Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
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, and a group is
imperfect The imperfect ( 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 English "was walking" or "used ...
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, $\,$ 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 and denoted with $G/N.$ There is a natural homomorphism, $f : G \to G/N,$ 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 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, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word iso ...
to $G / \ker f$ (the first isomorphism theorem). 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 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'' with respect to ''R'' ...
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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
$G.$

# 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 * 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)

# References

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

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