Normal subgroup

"Invariant subgroup" redirects here. Not to be confused withFully invariant subgroup.

Inabstract algebra,anormal subgroup(also known as aninvariant subgrouporself-conjugate subgroup)is asubgroupthat isinvariantunderconjugationby members of thegroupof which it is a part. In other words, a subgroup$N$of the group$G$is normal in$G$if and only if$gng^{-1}\in N$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 constructquotient groupsof the given group. Furthermore, the normal subgroups of$G$are precisely thekernelsofgroup homomorphismswithdomain$G$,which means that they can be used to internally classify those homomorphisms.

Évariste Galoiswas the first to realize the importance of the existence of normal subgroups.

Definitions

Asubgroup$N$of a group$G$is called anormal subgroupof$G$if it is invariant underconjugation;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 areequivalentto$N$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$N$,i.e.,$gNg^{-1}\subseteq N$for all$g\in G$.
• The image of conjugation of$N$by any element of$G$is equal to$N,$i.e.,$gNg^{-1}=N$for all$g\in G$.
• For all$g\in G$,the left and right cosets$gN$and$Ng$are equal.
• The sets of left and rightcosetsof$N$in$G$coincide.
• Multiplication in$G$preserves the equivalence relation "is in the same left coset as". That is, for every$g,g',h,h'\in G$satisfying$gN=g'N$and$hN=h'N$,we have$(gh)N=(g'h')N$.
• There exists a group on the set of left cosets of$N$where multiplication of any two left cosets$gN$and$hN$yields the left coset$(gh)N$(this group is called thequotient groupof$G$modulo$N$,denoted$G/N$).
• $N$is aunionofconjugacy classesof$G$.
• $N$is preserved by theinner automorphismsof$G$.
• There is somegroup homomorphism$G\to H$whosekernelis$N$.
• There exists a group homomorphism$\phi:G\to H$whose fibers form a group where the identity element is$N$and multiplication of any two fibers$\phi ^{-1}(h_{1})$and$\phi ^{-1}(h_{2})$yields the fiber$\phi ^{-1}(h_{1}h_{2})$(this group is the same group$G/N$mentioned above).
• There is somecongruence relationon$G$for which theequivalence classof theidentity elementis$N$.
• For all$n\in N$and$g\in G$.thecommutator$[n,g]=n^{-1}g^{-1}ng$is in$N$.
• Any two elements commute modulo the normal subgroup membership relation. That is, for all$g,h\in G$,$gh\in N$if and only if$hg\in N$.

Examples

For any group$G$,the trivial subgroup$\{e\}$consisting of only 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).Other named normal subgroups of an arbitrary group include thecenter of the group(the set of elements that commute with all other elements) and thecommutator subgroup$[G,G]$.More generally, since conjugation is an isomorphism, anycharacteristic subgroupis a normal subgroup.

If$G$is anabelian groupthen every subgroup$N$of$G$is normal, because$gN=\{gn\}_{n\in N}=\{ng\}_{n\in N}=Ng$.More generally, for any group$G$,every subgroup of thecenter$Z(G)$of$G$is normal in$G$(in the special case that$G$is abelian, the center is all of$G$,hence the fact that all subgroups of an abelian group are normal). A group that is not abelian but for which every subgroup is normal is called aHamiltonian group.

A concrete example of a normal subgroup is the subgroup$N=\{(1),(123),(132)\}$of thesymmetric 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$(12)N=\{(12),(23),(13)\}$.On the other hand, the subgroup$H=\{(1),(12)\}$is not normal in$S_{3}$since$(123)H=\{(123),(13)\}\neq \{(123),(23)\}=H(123)$.This illustrates the general fact that any subgroup$H\leq G$of index two is normal.

As an example of a normal subgroup within amatrix group,consider thegeneral linear group$\mathrm {GL} _{n}(\mathbf {R} )$of all invertible$n\times n$matrices with real entries under the operation of matrix multiplication and its subgroup$\mathrm {SL} _{n}(\mathbf {R} )$of all$n\times n$matrices ofdeterminant1 (thespecial linear group). To see why the subgroup$\mathrm {SL} _{n}(\mathbf {R} )$is normal in$\mathrm {GL} _{n}(\mathbf {R} )$,consider any matrix$X$in$\mathrm {SL} _{n}(\mathbf {R} )$and any invertible matrix$A$.Then using the two important identities$\det(AB)=\det(A)\det(B)$and$\det(A^{-1})=\det(A)^{-1}$,one has that$\det(AXA^{-1})=\det(A)\det(X)\det(A)^{-1}=\det(X)=1$,and so$AXA^{-1}\in \mathrm {SL} _{n}(\mathbf {R} )$as well. This means$\mathrm {SL} _{n}(\mathbf {R} )$is closed under conjugation in$\mathrm {GL} _{n}(\mathbf {R} )$,so it is a normal subgroup.

In theRubik's Cube group,the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.

Thetranslation groupis a normal subgroup of theEuclidean groupin 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 allrotationsabout the origin isnota 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 atransitive relation.The smallest group exhibiting this phenomenon is thedihedral groupof order 8.However, acharacteristic subgroupof a normal subgroup is normal.A group in which normality is transitive is called aT-group.
• The two groups$G$and$H$are normal subgroups of theirdirect product$G\times H$.
• If the group$G$is asemidirect product$G=N\rtimes H$,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$M\cap N=\{0\}$,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$f(N)$is normal in$H$.
• Normality is preserved by takinginverse images;that is, if$G\to H$is a group homomorphism and$N$is normal in$H$,then the inverse image$f^{-1}(N)$is normal in$G$.
• Normality is preserved on takingdirect 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 ofindex2 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 thenormal 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 issimpleif and only if it is isomorphic to all of its non-identity homomorphic images,a finite group isperfectif and only if it has no normal subgroups of primeindex,and a group isimperfectif and only if thederived subgroupis 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$NM=\{nm:n\in N\;{\text{ and }}\;m\in M\}$are also normal subgroups of$G$.

The normal subgroups of$G$form alatticeundersubset inclusionwithleast element,$\{e\}$,andgreatest element,$G$.Themeetof two normal subgroups,$N$and$M$,in this lattice is their intersection and thejoinis their product.

The lattice iscompleteandmodular.

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$a_{1}'\in a_{1}N,a_{2}'\in a_{2}N$.Then there are$n_{1},n_{2}\in N$such that$a_{1}'=a_{1}n_{1},a_{2}'=a_{2}n_{2}$.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 anormalsubgroup, and therefore there is$n_{1}'\in N$such that$n_{1}a_{2}=a_{2}n_{1}'$.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 groupand denoted with$G/N.$There is a naturalhomomorphism,$f:G\to G/N$,given by$f(a)=aN$.This homomorphism maps$N$into the identity element of$G/N$,which is the coset$eN=N$,that is,$\ker(f)=N$.

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$\{e\}$in$H$thekernelof the homomorphism and denote it by$\ker f$.As it turns out, the kernel is always normal and the image of$G,f(G)$,is alwaysisomorphicto$G/\ker f$(thefirst 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 toisomorphism).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 withdomain$G$.

See also

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)
• Semigroup ideal

Bibliography

• Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010)."On Rubik's Cube"(PDF).KTH.
• Cantrell, C.D. (2000).Modern Mathematical Methods for Physicists and Engineers.Cambridge University Press.ISBN978-0-521-59180-5.
• Dõmõsi, Pál; Nehaniv, Chrystopher L. (2004).Algebraic Theory of Automata Networks.SIAM Monographs on Discrete Mathematics and Applications. SIAM.
• Dummit, David S.; Foote, Richard M. (2004).Abstract Algebra(3rd ed.). John Wiley & Sons.ISBN0-471-43334-9.
• Fraleigh, John B. (2003).A First Course in Abstract Algebra(7th ed.). Addison-Wesley.ISBN978-0-321-15608-2.
• Hall, Marshall (1999).The Theory of Groups.Providence: Chelsea Publishing.ISBN978-0-8218-1967-8.
• Hungerford, Thomas (2003).Algebra.Graduate Texts in Mathematics. Springer.
• Hungerford, Thomas (2013).Abstract Algebra: An Introduction.Brooks/Cole Cengage Learning.
• Judson, Thomas W. (2020).Abstract Algebra: Theory and Applications.
• Robinson, Derek J. S. (1996).A Course in the Theory of Groups.Graduate Texts in Mathematics. Vol. 80 (2nd ed.).Springer-Verlag.ISBN978-1-4612-6443-9.Zbl0836.20001.
• Thurston, William(1997). Levy, Silvio (ed.).Three-dimensional geometry and topology, Vol. 1.Princeton Mathematical Series. Princeton University Press.ISBN978-0-691-08304-9.
• Bradley, C. J. (2010).The mathematical theory of symmetry in solids: representation theory for point groups and space groups.Oxford New York: Clarendon Press.ISBN978-0-19-958258-7.OCLC859155300.

Further reading

• I. N. Herstein,Topics in algebra.Second edition. Xerox College Publishing, Le xing ton, Mass.-Toronto, Ont., 1975. xi+388 pp.

External links

• Weisstein, Eric W."normal subgroup".MathWorld.
• Normal subgroup in Springer's Encyclopedia of Mathematics
• Robert Ash: Group Fundamentals inAbstract Algebra. The Basic Graduate Year
• Timothy Gowers, Normal subgroups and quotient groups
• John Baez, What's a Normal Subgroup?