Normal subgroup

In abstract algebra, a normal subgroup is a subgroup 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 $gng -1 \in N$ 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 construct quotient groups 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 was the first to realize the importance of the existence of normal subgroups.[1]

Definitions

A subgroup $N$ 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 $N$ .[2] The usual notation for this relation is $N\triangleleft G$ , and the definition may be written in symbols as

N\triangleleft G\Leftrightarrow \forall n\in N,\forall g\in G\colon gng^{-1}\in N.

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:

Any two elements commute regarding the normal subgroup membership relation: $\forall g, h \in G, gh \in N \Leftrightarrow hg \in N$ .
The image of conjugation of $N$ by any element of $G$ is a subset of $N$ : $\forall g \in G, gNg -1 \subseteq N$ .[3]'
The image of conjugation of $N$ by any element of $G$ is $N$ : $\forall g \in G, gNg -1 = N$ .[3]
$\forall g \in G, gN = Ng$ .[3]
The sets of left and right cosets of $N$ in $G$ coincide.[3]
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 $gh$ : $\forall x, y, g, h \in G$ , if $x \in gN$ and $y \in hN$ then $xy \in (gh)N$ .
$N$ is a union of conjugacy classes of $G$ : $N = ⋃ g \in N Cl(g)$ .[1]
$N$ is preserved by inner automorphisms.[4]
For all $n\in N$ and $g\in G$ , the commutator $[n,g]=n^{-1}g^{-1}ng$ is in $N$ .
There is some group homomorphism $G \to H$ whose kernel is $N$ .[1]

Examples

The trivial subgroup ${e}$ consisting of just the identity element of $G$ and $G$ itself are always normal subgroups of $G$ . If these are the only normal subgroups, then $G$ is said to be simple.[5]
The translation group is a normal subgroup of the Euclidean group in any dimension.[6] This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation (though typically a different one than the one we used earlier). By contrast, the subgroup of all 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.
Every subgroup $N$ of an abelian group $G$ is normal, because $gN = Ng$ . A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.[7]
The center of a group is a normal subgroup.[8]
More generally, any characteristic subgroup is normal, since conjugation is always an automorphism.[9]
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.[10]
The commutator subgroup $[G,G]$ is a normal subgroup of $G$ [11]. Given elements $g\in G$ and $h\in [G,G]$ , the commutator of these two elements is $[g,h]=ghg^{-1}h^{-1}$ . Now, because the commutator is closed under multiplication $(ghg^{-1}h^{-1})h\in [G,G]$ and $(ghg^{-1}h^{-1})h=ghg^{-1}$ , the commutator is a normal subgroup.

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 $K$ .[12]
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.[13] However, a characteristic subgroup of a normal subgroup is normal.[14] A group in which normality is transitive is called a T-group.[15]
The two groups $G$ and $H$ are normal subgroups of their direct product $G \times H$ .
If the group $G$ is a semidirect product $G=N\rtimes H,$ , then $N$ is normal in $G$ , though $H$ need not be normal in $G$ .
Normality is preserved under surjective homomorphisms,[16] i.e. 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 taking inverse images,[16] i.e. 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 taking direct products,[17] i.e. 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.[18]
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,[19] a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is 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 $NM=\{nm\mid n\in N\;{\text{ and }}\;m\in M\}$ are also normal subgroups of $G$ .

The normal subgroups of $G$ form a lattice under subset inclusion with least element, ${e}$ , 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.[17]

Normal subgroups, quotient groups and homomorphisms

If $N$ is a normal subgroup, we can define a multiplication on cosets as follows:

(a_{1}N)(a_{2}N):=(a_{1}a_{2})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 a normal subgroup, 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 the quotient group and denoted with $G / N$ . There is a natural homomorphism, $f : G \to G/N$ , given by $f (a) = aN$ . This homomorphism maps $N$ into the identity element of $G/N$ , which is the coset $eN = N$ ,[20] 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 $H$ the kernel of the homomorphism and denote it by $ker(f)$ . As it turns out, the kernel is always normal and the image of $G$ , $f (G)$ , is always isomorphic to $G /ker(f)$ (the first isomorphism theorem).[21] 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).[22] 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 $G$ .[23]

Notes

Cantrell 2000, p. 160.
Dummit & Foote 2004.
Hungerford 2003, p. 41.
Fraleigh 2003, p. 141.
Robinson 1996, p. 16.
Thurston 1997, p. 218.
Hall 1999, p. 190.
Hungerford 2003, p. 45.
Hall 1999, p. 32.
Bergvall et al. 2010, p. 96.
Hall 1999, p. 138.
Hungerford 2003, p. 42.
Robinson 1996, p. 17.
Robinson 1996, p. 28.
Robinson 1996, p. 402.
Hall 1999, p. 29.
Hungerford 2003, p. 46.
Robinson 1996, p. 36.
Dõmõsi & Nehaniv 2004, p. 7.
Hungerford 2003, pp. 42–43.
Hungerford 2003, p. 44.
Robinson 1996, p. 20.
Hall 1999, p. 27.

References

Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010). "On Rubik's Cube" (PDF). KTH. Cite journal requires |journal= (help)CS1 maint: ref=harv (link)
Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 978-0-521-59180-5.CS1 maint: ref=harv (link)
Dõmõsi, Pál; Nehaniv, Chrystopher L. (2004). Algebraic Theory of Automata Networks. SIAM Monographs on Discrete Mathematics and Applications. SIAM.CS1 maint: ref=harv (link)
Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.CS1 maint: ref=harv (link)
Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley. ISBN 978-0-321-15608-2.CS1 maint: ref=harv (link)
Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8.CS1 maint: ref=harv (link)
Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer.CS1 maint: ref=harv (link)
Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001.CS1 maint: ref=harv (link)
Thurston, William (1997). Levy, Silvio (ed.). Three-dimensional geometry and topology, Vol. 1. Princeton Mathematical Series. Princeton University Press. ISBN 978-0-691-08304-9.CS1 maint: ref=harv (link)

External links

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[FOOTNOTECantrell2000160-1] Cantrell 2000, p. 160.

[FOOTNOTEDummitFoote2004-2] Dummit & Foote 2004.

[FOOTNOTEHungerford200341-3] Hungerford 2003, p. 41.

[FOOTNOTEFraleigh2003141-4] Fraleigh 2003, p. 141.

[FOOTNOTERobinson199616-5] Robinson 1996, p. 16.

[FOOTNOTEThurston1997218-6] Thurston 1997, p. 218.

[FOOTNOTEHall1999190-7] Hall 1999, p. 190.

[FOOTNOTEHungerford200345-8] Hungerford 2003, p. 45.

[FOOTNOTEHall199932-9] Hall 1999, p. 32.

[FOOTNOTEBergvallHynningHedbergMickelin201096-10] Bergvall et al. 2010, p. 96.

[FOOTNOTEHall1999138-11] Hall 1999, p. 138.

[FOOTNOTEHungerford200342-12] Hungerford 2003, p. 42.

[FOOTNOTERobinson199617-13] Robinson 1996, p. 17.

[FOOTNOTERobinson199628-14] Robinson 1996, p. 28.

[FOOTNOTERobinson1996402-15] Robinson 1996, p. 402.

[FOOTNOTEHall199929-16] Hall 1999, p. 29.

[FOOTNOTEHungerford200346-17] Hungerford 2003, p. 46.

[FOOTNOTERobinson199636-18] Robinson 1996, p. 36.

[FOOTNOTEDõmõsiNehaniv20047-19] Dõmõsi & Nehaniv 2004, p. 7.

[FOOTNOTEHungerford200342–43-20] Hungerford 2003, pp. 42–43.

[FOOTNOTEHungerford200344-21] Hungerford 2003, p. 44.

[FOOTNOTERobinson199620-22] Robinson 1996, p. 20.

[FOOTNOTEHall199927-23] Hall 1999, p. 27.

Normal subgroup

Definitions

Equivalent conditions

Examples

Properties

Lattice of normal subgroups

Normal subgroups, quotient groups and homomorphisms

See also

Operations taking subgroups to subgroups

Subgroup properties complementary (or opposite) to normality

Subgroup properties stronger than normality

Subgroup properties weaker than normality

Related notions in algebra

Notes

References

Further reading

External links