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 $H$ of a group $G$ is normal in $G$ if and only if $gH = Hg$ for all $g$ in $G$ . The definition of normal subgroup implies that the sets of left and right cosets coincide. In fact, a seemingly weaker condition that the sets of left and right cosets coincide also implies that the subgroup $H$ of a group $G$ is normal in $G$ . Normal subgroups (and only normal subgroups) can be used to construct quotient groups from a given group.

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 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]

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

For any subgroup, the following conditions are equivalent to normality. 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 $gh$ : $\forall x, y, g, h \in G, x \in gN and y \in hN \Rightarrow xy \in (gh)N$ .
$N$ is a union of conjugacy classes of $G$ : $N = ⋃ g \in N Cl(g)$ .^[1]
$N$ is preserved by inner automorphisms.^[4]
There is some homomorphism on $G$ for which $N$ is the kernel: $\existsφ \in Hom(G) ∣ ker φ = N$ .^[1]

The last condition 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,^[5] 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.

Examples

The subgroup ${e}$ consisting of just the identity element of $G$ and $G$ itself are always normal subgroups of $G$ . The former is called the trivial subgroup, and if these are the only normal subgroups, then $G$ is said to be simple.^[6]
The center of a group is a normal subgroup.^[7]
The commutator subgroup is a normal subgroup.^[8]
More generally, any characteristic subgroup is normal, since conjugation is always an automorphism.^[9]
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.^[10]
The translation group is a normal subgroup of the Euclidean group in any dimension.^[11]
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.^[12]

Properties

Normality is preserved upon surjective homomorphisms, and is also preserved upon taking inverse images.^[13]
Normality is preserved on taking direct products.^[14]
If $H$ is a normal subgroup of $G$ , and $K$ is a subgroup of $G$ containing $H$ , then $H$ is a normal subgroup of $K$ .^[15]
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.^[16] However, a characteristic subgroup of a normal subgroup is normal.^[17] A group in which normality is transitive is called a T-group.^[18]
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.^[19]

Lattice of normal subgroups

The normal subgroups of a group, $G$ , form a lattice under subset inclusion with least element, ${e}$ , and greatest element, $G$ . Given two normal subgroups, $N$ and $M$ , in $G$ , meet is defined as

N\wedge M:=N\cap M

and join is defined as

N\vee M:=NM=\{nm\,|\,n\in N{\text{, and }}m\in M\}.

The lattice is complete and modular.^[14]

Normal subgroups 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.

With this operation, the set of cosets is itself a group, called the quotient group and denoted $G / N$ . There is a natural homomorphism, $f : G \to G/N$ given by $f (a) = aN$ . The image $f (N)$ consists only of the identity element of $G/N$ , the coset $eN = N$ .^[20]

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

^ ^a ^b ^c Cantrell 2000, p. 160.
^ Dummit & Foote 2004.
^ ^a ^b ^c ^d Hungerford 2003, p. 41.
^ Fraleigh 2003, p. 141.
^ Dõmõsi & Nehaniv 2004, p. 7.
^ Robinson 1996, p. 16.
^ Hungerford 2003, p. 45.
^ Hall 1999, p. 138.
^ Hall 1999, p. 32.
^ Hall 1999, p. 190.
^ Thurston 1997, p. 218.
^ Bergvall et al. 2010, p. 96.
^ Hall 1999, p. 29.
^ ^a ^b Hungerford 2003, p. 46.
^ Hungerford 2003, p. 42.
^ Robinson 1996, p. 17.
^ Robinson 1996, p. 28.
^ Robinson 1996, p. 402.
^ Robinson 1996, p. 36.
^ 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}}: Cite journal requires |journal= (help); Invalid |ref=harv (help)
Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 978-0-521-59180-5. {{cite book}}: Invalid |ref=harv (help)
Dõmõsi, Pál; Nehaniv, Chrystopher L. (2004). Algebraic Theory of Automata Networks. SIAM Monographs on Discrete Mathematics and Applications. SIAM. {{cite book}}: Invalid |ref=harv (help)
Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. {{cite book}}: Invalid |ref=harv (help)
Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley. ISBN 978-0-321-15608-2. {{cite book}}: Invalid |ref=harv (help)
Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8. {{cite book}}: Invalid |ref=harv (help)
Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer. {{cite book}}: Invalid |ref=harv (help)
Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001. {{cite book}}: Invalid |ref=harv (help)
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. {{cite book}}: Invalid |ref=harv (help)

External links

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

[FOOTNOTEDummitFoote2004-2] Dummit & Foote 2004.

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

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

[FOOTNOTEDõmõsiNehaniv20047-5] Dõmõsi & Nehaniv 2004, p. 7.

[FOOTNOTERobinson199616-6] Robinson 1996, p. 16.

[FOOTNOTEHungerford200345-7] Hungerford 2003, p. 45.

[FOOTNOTEHall1999138-8] Hall 1999, p. 138.

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

[FOOTNOTEHall1999190-10] Hall 1999, p. 190.

[FOOTNOTEThurston1997218-11] Thurston 1997, p. 218.

[FOOTNOTEBergvallHynningHedbergMickelin201096-12] Bergvall et al. 2010, p. 96.

[FOOTNOTEHall199929-13] Hall 1999, p. 29.

[FOOTNOTEHungerford200346-14] Hungerford 2003, p. 46.

[FOOTNOTEHungerford200342-15] Hungerford 2003, p. 42.

[FOOTNOTERobinson199617-16] Robinson 1996, p. 17.

[FOOTNOTERobinson199628-17] Robinson 1996, p. 28.

[FOOTNOTERobinson1996402-18] Robinson 1996, p. 402.

[FOOTNOTERobinson199636-19] Robinson 1996, p. 36.

[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.

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Normal subgroup

Definitions

Examples

Properties

Lattice of normal subgroups

Normal subgroups 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