In

Normal subgroup in Springer's Encyclopedia of Mathematics

Robert Ash: Group Fundamentals in ''Abstract Algebra. The Basic Graduate Year''

Timothy Gowers, Normal subgroups and quotient groups

Subgroup properties

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
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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 $gng^\; \backslash in\; N$ for all $g\; \backslash in\; G$ and $n\; \backslash in\; N.$ The usual notation for this relation is $N\; \backslash triangleleft\; G.$
Normal subgroups are important because they (and only they) can be used to construct quotient group
A quotient group or factor group is a math
Mathematics (from Greek: ) includes the study of such topics as quantity ( number theory), structure (algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit= ...

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 Ancient Greek, Greek: ) includes the study of such topics as qu ...

was the first to realize the importance of the existence of normal subgroups.
Definitions

Asubgroup
In group theory
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$N$ 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 $N\; \backslash triangleleft\; G.$
Equivalent conditions

For any subgroup $N$ of $G,$ the following conditions areequivalent
Equivalence or Equivalent may refer to:
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*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 $g\; \backslash in\; G,$ 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 $x,\; y,\; g,\; h\; \backslash in\; G,$ if $x\; \backslash in\; g\; N$and $y\; \backslash in\; h\; N$ then $x\; y\; \backslash in\; (g\; h)\; N.$
* $N$ is a union of conjugacy class
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). It ...

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
Image: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 ...

$G\; \backslash to\; H$ whose kernel
Kernel may refer to:
Computing
* Kernel (operating system)
In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...

is $N.$
* For all $n\backslash in\; N$ and $g\backslash in\; G,$ the commutator
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). It ...

$;\; href="/html/ALL/s/,g.html"\; ;"title=",g">,g$ is in $N.$
* Any two elements commute regarding the normal subgroup membership relation: for all $g,\; h\; \backslash in\; G,$ $g\; h\; \backslash in\; N$ if and only if $h\; g\; \backslash in\; N.$
Examples

For any group $G,$ the trivial subgroup $\backslash $ 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 besimple
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
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). It h ...

$;\; href="/html/ALL/s/,G.html"\; ;"title=",G">,G$ More generally, since conjugation is an isomorphism, any characteristic subgroup
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). It h ...

is a normal subgroup.
If $G$ is an abelian group
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 ( and ). There is no g ...

then every subgroup $N$ of $G$ is normal, because $gN\; =\; \backslash \_\; =\; \backslash \_\; =\; Ng.$ 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\; =\; \backslash $ of the symmetric group $S\_3,$ 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 $(12)N\; =\; \backslash .$ On the other hand, the subgroup $H\; =\; \backslash $ is not normal in $S\_3$ since $(123)H\; =\; \backslash \; \backslash neq\; \backslash \; =\; H(123).$ This illustrates the general fact that any subgroup $H\; \backslash leq\; G$ 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 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 Rotation, rotations 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. The smallest group exhibiting this phenomenon is the dihedral group of order 8. However, acharacteristic subgroup
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). It h ...

of a normal subgroup is normal. A group in which normality is transitive is called a T-group (mathematics), T-group.
* The two groups $G$ and $H$ are normal subgroups of their Direct product of groups, direct product $G\; \backslash times\; H.$
* If the group $G$ is a semidirect product $G\; =\; N\; \backslash rtimes\; H,$ then $N$ is normal in $G,$ though $H$ need not be normal in $G.$
* Normality is preserved under surjective homomorphisms; that is, if $G\; \backslash to\; H$ is a surjective group homomorphism and $N$ is normal in $G,$ then the image $f(N)$ is normal in $H.$
* Normality is preserved by taking Inverse image, inverse images; that is, if $G\; \backslash to\; H$ is a group homomorphism and $N$ is normal in $H,$ then the inverse image $f^(N)$ is normal in $G.$
* Normality is preserved on taking direct product of groups, direct products; that is, if $N\_1\; \backslash triangleleft\; G\_1$ and $N\_2\; \backslash triangleleft\; G\_2,$ then $N\_1\; \backslash times\; N\_2\backslash ;\; \backslash triangleleft\; \backslash ;G\_1\; \backslash times\; G\_2.$
* Every subgroup of Index (group theory), 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 group, 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 $N\backslash cap\; M$and their product $N\; M\; =\; \backslash $ are also normal subgroups of $G.$ The normal subgroups of $G$ form a Lattice (order), lattice under subset inclusion with least element, $\backslash ,$ 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: $$\backslash left(a\_1\; N\backslash right)\; \backslash left(a\_2\; N\backslash right)\; :=\; \backslash left(a\_1\; a\_2\backslash right)\; N.$$ This relation defines a mapping $G/N\backslash times\; G/N\; \backslash to\; G/N.$ 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 $a\_1\text{'}\backslash in\; a\_1\; N,\; a\_2\text{'}\; \backslash in\; a\_2\; N.$ Then there are $n\_1,\; n\_2\backslash in\; N$ such that $a\_1\text{'}\; =\; a\_1\; n\_1,\; a\_2\text{'}\; =\; a\_2\; n\_2.$ It follows that $$a\_1\text{'}\; a\_2\text{'}\; N\; =\; a\_1\; n\_1\; a\_2\; n\_2\; N\; =a\_1\; a\_2\; n\_1\text{'}\; n\_2\; N=a\_1\; a\_2\; N,$$where we also used the fact that $N$ is a subgroup, and therefore there is $n\_1\text{'}\backslash in\; N$ 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 thequotient group
A quotient group or factor group is a math
Mathematics (from Greek: ) includes the study of such topics as quantity ( number theory), structure (algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit= ...

and denoted with $G/N.$ There is a natural Group homomorphism, homomorphism, $f\; :\; G\; \backslash to\; G/N,$ given by $f(a)\; =\; a\; N.$ This homomorphism maps $N$ into the identity element of $G/N,$ which is the coset $e\; N\; =\; N,$ that is, $\backslash ker(f)\; =\; N.$
In general, a group homomorphism, $f\; :\; G\; \backslash to\; H$ 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 $\backslash $ in $H$ the kernel
Kernel may refer to:
Computing
* Kernel (operating system)
In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...

of the homomorphism and denote it by $\backslash ker\; f.$ As it turns out, the kernel is always normal and the image of $G,\; f(G),$ is always isomorphic to $G\; /\; \backslash ker\; f$ (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, $f\; :\; G\; \backslash to\; G/N,$ is $N$ itself, so the normal subgroups are precisely the kernels of homomorphisms with Domain of a function, domain $G.$
See also

Operations taking subgroups to subgroups

*Normalizer *Conjugate closure *Normal coreSubgroup properties complementary (or opposite) to normality

*Malnormal subgroup *Contranormal subgroup *Abnormal subgroup *Self-normalizing subgroupSubgroup properties stronger than normality

*Characteristic subgroup *Fully characteristic subgroupSubgroup 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 subgroupRelated notions in algebra

*Ideal (ring theory)Notes

References

* * * * * * * * * * *Further reading

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

* {{MathWorld, urlname=NormalSubgroup, title= normal subgroupNormal subgroup in Springer's Encyclopedia of Mathematics

Robert Ash: Group Fundamentals in ''Abstract Algebra. The Basic Graduate Year''

Timothy Gowers, Normal subgroups and quotient groups

Subgroup properties