normal subgroup

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In
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a
subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
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 A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
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 "factore ...
s of the given group. Furthermore, the normal subgroups of $G$ are precisely the kernels of
group homomorphismsImage:Group homomorphism ver.2.svg, 250px, Image of a group homomorphism (h) from G (left) to H (right). The smaller oval inside H is the image of h. N is the Kernel_(algebra)#Group_homomorphisms, kernel of h and aN is a coset of N. In mathematics, ...

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 A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...
was the first to realize the importance of the existence of normal subgroups.

# Definitions

A
subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
$N$ 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 *''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
coset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s 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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
es of $G.$ * $N$ is preserved by the
inner automorphism In abstract algebra an inner automorphism is an automorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...
s of $G.$ * There is some
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

$G \to H$ 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 $N.$ * For all $n\in N$ and $g\in G,$ the
commutator In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is in $N.$ * Any two elements commute regarding the normal subgroup membership relation: 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), ''Simple'' (album), by Andy Yorke, 2008, and its title track * Simple (Florida Georgia Line song), "Simple" (Florida Ge ...
.) 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
More generally, since conjugation is an isomorphism, any
characteristic subgroup In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
is a normal subgroup. If $G$ is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 In group theory, a Dedekind group is a group (mathematics), group ''G'' such that every subgroup of ''G'' is normal subgroup, normal. All abelian groups are Dedekind groups. A non-abelian Dedekind group is called a Hamiltonian group. The most famil ...
. A concrete example of a normal subgroup is the subgroup $N = \$ of the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
$S_3,$ 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 Rubik's Cube group is a Group (mathematics), group (G, \cdot ) that represents the Mathematical structure, structure of the Rubik's Cube mechanical puzzle. Each element of the Set (mathematics), set G corresponds to a cube move, which is the e ...
, 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 geometry, distance in a given direction. A translation can also be interpreted as the addition of a constan ...
is a normal subgroup of the
Euclidean group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 is the circular movement of an object around an ''axis of rotation''. A three-dimensional object may have an infinite number of rotation axes. If the rotation axis passes internally through the body's own center of mass, then the bo ...

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 (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. The smallest group exhibiting this phenomenon is the
dihedral group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of order 8. However, a
characteristic subgroup In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
$G \times H.$ * If the group $G$ is a
semidirect product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
$G = N \rtimes H,$ then $N$ is normal in $G,$ though $H$ need not be normal in $G.$ * 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 ; 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 group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group (mathematics), group. The two most common types are the normal core of a subgroup and the ''p''-core of a group. The normal core Definition For a ...
. 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), ''Simple'' (album), by Andy Yorke, 2008, and its title track * Simple (Florida Georgia Line song), "Simple" (Florida Ge ...
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 of a subgroup, index, and a group is Imperfect group, imperfect 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 (order), lattice under subset inclusion with least element, $\,$ and greatest element, $G.$ The Meet (lattice theory), meet of two normal subgroups, $N$ and $M,$ in this lattice is their intersection and the Join (lattice theory), join is their product. The lattice is Complete lattice, complete and Modular lattice, modular.

# 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 "factore ...
and denoted with $G/N.$ There is a natural Group homomorphism, 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 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 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 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 of a function, domain $G.$

## 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.