In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup)[1] is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup ${\displaystyle N}$ of the group ${\displaystyle G}$ is normal in ${\displaystyle G}$ if and only if ${\displaystyle gng^{-1}\in N}$ for all ${\displaystyle g\in G}$ and ${\displaystyle n\in N.}$ The usual notation for this relation is ${\displaystyle N\triangleleft G.}$

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 ${\displaystyle G}$ are precisely the kernels of group homomorphisms with domain ${\displaystyle G,}$ 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.[2]

## Definitions

A subgroup ${\displaystyle N}$ of a group ${\displaystyle G}$ is called a normal subgroup of ${\displaystyle G}$ if it is invariant under conjugation; that is, the conjugation of an element of ${\displaystyle N}$ by an element of ${\displaystyle G}$ is always in ${\displaystyle N.}$[3] The usual notation for this relation is ${\displaystyle N\triangleleft G.}$

### Equivalent conditions

For any subgroup ${\displaystyle N}$ of ${\displaystyle G,}$ the following conditions are equivalent to ${\displaystyle N}$ being a normal subgroup of ${\displaystyle G.}$ Therefore, any one of them may be taken as the definition.

• The image of conjugation of ${\displaystyle N}$ by any element of ${\displaystyle G}$ is a subset of ${\displaystyle N,}$[4] i.e., ${\displaystyle gNg^{-1}\subseteq N}$ for all ${\displaystyle g\in G}$.
• The image of conjugation of ${\displaystyle N}$ by any element of ${\displaystyle G}$ is equal to ${\displaystyle N,}$[4] i.e., ${\displaystyle gNg^{-1}=N}$ for all ${\displaystyle g\in G}$.
• For all ${\displaystyle g\in G,}$ the left and right cosets ${\displaystyle gN}$ and ${\displaystyle Ng}$ are equal.[4]
• The sets of left and right cosets of ${\displaystyle N}$ in ${\displaystyle G}$ coincide.[4]
• Multiplication in ${\displaystyle G}$ preserves the equivalence relation "is in the same left coset as". That is, for every ${\displaystyle g,g',h,h'\in G}$ satisfying ${\displaystyle gN=g'N}$ and ${\displaystyle hN=h'N}$, we have ${\displaystyle (gh)N=(g'h')N.}$
• There exists a group on the set of left cosets of ${\displaystyle N}$ where multiplication of any two left cosets ${\displaystyle gN}$ and ${\displaystyle hN}$ yields the left coset ${\displaystyle (gh)N}$. (This group is called the quotient group of ${\displaystyle G}$ modulo ${\displaystyle N}$, denoted ${\displaystyle G/N}$.)
• ${\displaystyle N}$ is a union of conjugacy classes of ${\displaystyle G.}$[2]
• ${\displaystyle N}$ is preserved by the inner automorphisms of ${\displaystyle G.}$[5]
• There is some group homomorphism ${\displaystyle G\to H}$ whose kernel is ${\displaystyle N.}$[2]
• There exists a group homomorphism ${\displaystyle \phi :G\to H}$ whose fibers form a group where the identity element is ${\displaystyle N}$ and multiplication of any two fibers ${\displaystyle \phi ^{-1}(h_{1})}$ and ${\displaystyle \phi ^{-1}(h_{2})}$ yields the fiber ${\displaystyle \phi ^{-1}(h_{1}h_{2})}$. (This group is the same group ${\displaystyle G/N}$ mentioned above.)
• There is some congruence relation on ${\displaystyle G}$ for which the equivalence class of the identity element is ${\displaystyle N}$.
• For all ${\displaystyle n\in N}$ and ${\displaystyle g\in G,}$ the commutator ${\displaystyle [n,g]=n^{-1}g^{-1}ng}$ is in ${\displaystyle N.}$[citation needed]
• Any two elements commute modulo the normal subgroup membership relation. That is, for all ${\displaystyle g,h\in G,}$ ${\displaystyle gh\in N}$ if and only if ${\displaystyle hg\in N.}$[citation needed]

## Examples

For any group ${\displaystyle G,}$ the trivial subgroup ${\displaystyle \{e\))$ consisting of just the identity element of ${\displaystyle G}$ is always a normal subgroup of ${\displaystyle G.}$ Likewise, ${\displaystyle G}$ itself is always a normal subgroup of ${\displaystyle G.}$ (If these are the only normal subgroups, then ${\displaystyle G}$ is said to be simple.)[6] 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 ${\displaystyle [G,G].}$[7][8] More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup.[9]

If ${\displaystyle G}$ is an abelian group then every subgroup ${\displaystyle N}$ of ${\displaystyle G}$ is normal, because ${\displaystyle gN=\{gn\}_{n\in N}=\{ng\}_{n\in N}=Ng.}$ More generally, for any group ${\displaystyle G}$, every subgroup of the center ${\displaystyle Z(G)}$ of ${\displaystyle G}$ is normal in ${\displaystyle G}$. (In the special case that ${\displaystyle G}$ is abelian, the center is all of ${\displaystyle G}$, 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 a Hamiltonian group.[10]

A concrete example of a normal subgroup is the subgroup ${\displaystyle N=\{(1),(123),(132)\))$ of the symmetric group ${\displaystyle S_{3},}$ consisting of the identity and both three-cycles. In particular, one can check that every coset of ${\displaystyle N}$ is either equal to ${\displaystyle N}$ itself or is equal to ${\displaystyle (12)N=\{(12),(23),(13)\}.}$ On the other hand, the subgroup ${\displaystyle H=\{(1),(12)\))$ is not normal in ${\displaystyle S_{3))$ since ${\displaystyle (123)H=\{(123),(13)\}\neq \{(123),(23)\}=H(123).}$[11] This illustrates the general fact that any subgroup ${\displaystyle H\leq G}$ of index two is normal.

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

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]

The translation group is a normal subgroup of the Euclidean group in any dimension.[13] 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 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 ${\displaystyle H}$ is a normal subgroup of ${\displaystyle G,}$ and ${\displaystyle K}$ is a subgroup of ${\displaystyle G}$ containing ${\displaystyle H,}$ then ${\displaystyle H}$ is a normal subgroup of ${\displaystyle K.}$[14]
• 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.[15] However, a characteristic subgroup of a normal subgroup is normal.[16] A group in which normality is transitive is called a T-group.[17]
• The two groups ${\displaystyle G}$ and ${\displaystyle H}$ are normal subgroups of their direct product ${\displaystyle G\times H.}$
• If the group ${\displaystyle G}$ is a semidirect product ${\displaystyle G=N\rtimes H,}$ then ${\displaystyle N}$ is normal in ${\displaystyle G,}$ though ${\displaystyle H}$ need not be normal in ${\displaystyle G.}$
• If ${\displaystyle M}$ and ${\displaystyle N}$ are normal subgroups of an additive group ${\displaystyle G}$ such that ${\displaystyle G=M+N}$ and ${\displaystyle M\cap N=\{0\))$, then ${\displaystyle G=M\oplus N.}$[18]
• Normality is preserved under surjective homomorphisms;[19] that is, if ${\displaystyle G\to H}$ is a surjective group homomorphism and ${\displaystyle N}$ is normal in ${\displaystyle G,}$ then the image ${\displaystyle f(N)}$ is normal in ${\displaystyle H.}$
• Normality is preserved by taking inverse images;[19] that is, if ${\displaystyle G\to H}$ is a group homomorphism and ${\displaystyle N}$ is normal in ${\displaystyle H,}$ then the inverse image ${\displaystyle f^{-1}(N)}$ is normal in ${\displaystyle G.}$
• Normality is preserved on taking direct products;[20] that is, if ${\displaystyle N_{1}\triangleleft G_{1))$ and ${\displaystyle N_{2}\triangleleft G_{2},}$ then ${\displaystyle N_{1}\times N_{2}\;\triangleleft \;G_{1}\times G_{2}.}$
• Every subgroup of index 2 is normal. More generally, a subgroup, ${\displaystyle H,}$ of finite index, ${\displaystyle n,}$ in ${\displaystyle G}$ contains a subgroup, ${\displaystyle K,}$ normal in ${\displaystyle G}$ and of index dividing ${\displaystyle n!}$ called the normal core. In particular, if ${\displaystyle p}$ is the smallest prime dividing the order of ${\displaystyle G,}$ then every subgroup of index ${\displaystyle p}$ is normal.[21]
• The fact that normal subgroups of ${\displaystyle G}$ are precisely the kernels of group homomorphisms defined on ${\displaystyle G}$ 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,[22] 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, ${\displaystyle N}$ and ${\displaystyle M,}$ of ${\displaystyle G,}$ their intersection ${\displaystyle N\cap M}$and their product ${\displaystyle NM=\{nm:n\in N\;{\text{ and ))\;m\in M\))$ are also normal subgroups of ${\displaystyle G.}$

The normal subgroups of ${\displaystyle G}$ form a lattice under subset inclusion with least element, ${\displaystyle \{e\},}$ and greatest element, ${\displaystyle G.}$ The meet of two normal subgroups, ${\displaystyle N}$ and ${\displaystyle M,}$ in this lattice is their intersection and the join is their product.

The lattice is complete and modular.[20]

## Normal subgroups, quotient groups and homomorphisms

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

${\displaystyle \left(a_{1}N\right)\left(a_{2}N\right):=\left(a_{1}a_{2}\right)N.}$
This relation defines a mapping ${\displaystyle G/N\times G/N\to G/N.}$ To show that this mapping is well-defined, one needs to prove that the choice of representative elements ${\displaystyle a_{1},a_{2))$ does not affect the result. To this end, consider some other representative elements ${\displaystyle a_{1}'\in a_{1}N,a_{2}'\in a_{2}N.}$ Then there are ${\displaystyle n_{1},n_{2}\in N}$ such that ${\displaystyle a_{1}'=a_{1}n_{1},a_{2}'=a_{2}n_{2}.}$ It follows that
${\displaystyle a_{1}'a_{2}'N=a_{1}n_{1}a_{2}n_{2}N=a_{1}a_{2}n_{1}'n_{2}N=a_{1}a_{2}N,}$
where we also used the fact that ${\displaystyle N}$ is a normal subgroup, and therefore there is ${\displaystyle n_{1}'\in N}$ such that ${\displaystyle n_{1}a_{2}=a_{2}n_{1}'.}$ 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 ${\displaystyle G/N.}$ There is a natural homomorphism, ${\displaystyle f:G\to G/N,}$ given by ${\displaystyle f(a)=aN.}$ This homomorphism maps ${\displaystyle N}$ into the identity element of ${\displaystyle G/N,}$ which is the coset ${\displaystyle eN=N,}$[23] that is, ${\displaystyle \ker(f)=N.}$

In general, a group homomorphism, ${\displaystyle f:G\to H}$ sends subgroups of ${\displaystyle G}$ to subgroups of ${\displaystyle H.}$ Also, the preimage of any subgroup of ${\displaystyle H}$ is a subgroup of ${\displaystyle G.}$ We call the preimage of the trivial group ${\displaystyle \{e\))$ in ${\displaystyle H}$ the kernel of the homomorphism and denote it by ${\displaystyle \ker f.}$ As it turns out, the kernel is always normal and the image of ${\displaystyle G,f(G),}$ is always isomorphic to ${\displaystyle G/\ker f}$ (the first isomorphism theorem).[24] In fact, this correspondence is a bijection between the set of all quotient groups of ${\displaystyle G,G/N,}$ and the set of all homomorphic images of ${\displaystyle G}$ (up to isomorphism).[25] It is also easy to see that the kernel of the quotient map, ${\displaystyle f:G\to G/N,}$ is ${\displaystyle N}$ itself, so the normal subgroups are precisely the kernels of homomorphisms with domain ${\displaystyle G.}$[26]

## Notes

1. ^ In other language: ${\displaystyle \det }$ is a homomorphism from ${\displaystyle \mathrm {GL} _{n}(\mathbf {R} )}$ to the multiplicative subgroup ${\displaystyle \mathbf {R} ^{\times ))$, and ${\displaystyle \mathrm {SL} _{n}(\mathbf {R} )}$ is the kernel. Both arguments also work over the complex numbers, or indeed over an arbitrary field.

## References

1. ^ Bradley 2010, p. 12.
2. ^ a b c Cantrell 2000, p. 160.
3. ^
4. ^ a b c d Hungerford 2003, p. 41.
5. ^ Fraleigh 2003, p. 141.
6. ^ Robinson 1996, p. 16.
7. ^ Hungerford 2003, p. 45.
8. ^ Hall 1999, p. 138.
9. ^ Hall 1999, p. 32.
10. ^ Hall 1999, p. 190.
11. ^ Judson 2020, Section 10.1.
12. ^ Bergvall et al. 2010, p. 96.
13. ^ Thurston 1997, p. 218.
14. ^ Hungerford 2003, p. 42.
15. ^ Robinson 1996, p. 17.
16. ^ Robinson 1996, p. 28.
17. ^ Robinson 1996, p. 402.
18. ^ Hungerford 2013, p. 290.
19. ^ a b Hall 1999, p. 29.
20. ^ a b Hungerford 2003, p. 46.
21. ^ Robinson 1996, p. 36.
22. ^ Dõmõsi & Nehaniv 2004, p. 7.
23. ^ Hungerford 2003, pp. 42–43.
24. ^ Hungerford 2003, p. 44.
25. ^ Robinson 1996, p. 20.
26. ^ Hall 1999, p. 27.

## Bibliography

• 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)
• Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 978-0-521-59180-5.
• 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. ISBN 0-471-43334-9.
• Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley. ISBN 978-0-321-15608-2.
• Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8.
• 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. ISBN 978-1-4612-6443-9. Zbl 0836.20001.
• 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.
• Bradley, C. J. (2010). The mathematical theory of symmetry in solids : representation theory for point groups and space groups. Oxford New York: Clarendon Press. ISBN 978-0-19-958258-7. OCLC 859155300.