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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 N {\displaystyle N} of the group G {\displaystyle G} is normal in G {\displaystyle G} if and only if g n g − 1 ∈ N {\displaystyle gng^{-1}\in N} for all g ∈ G {\displaystyle g\in G} and n ∈ N . {\displaystyle n\in N.} The usual notation for this relation is N ◃ G . {\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 G {\displaystyle G} are precisely the kernels of group homomorphisms with domain G , {\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 N {\displaystyle N} of a group G {\displaystyle G} is called a normal subgroup of G {\displaystyle G} if it is invariant under conjugation; that is, the conjugation of an element of N {\displaystyle N} by an element of G {\displaystyle G} is always in N . {\displaystyle N.} ^[3] The usual notation for this relation is N ◃ G . {\displaystyle N\triangleleft G.}

Examples

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

If G {\displaystyle G} is an abelian group then every subgroup N {\displaystyle N} of G {\displaystyle G} is normal, because g N = { g n } n ∈ N = { n g } n ∈ N = N g . {\displaystyle gN=\{gn\}_{n\in N}=\{ng\}_{n\in N}=Ng.} More generally, for any group G {\displaystyle G} , every subgroup of the center Z ( G ) {\displaystyle Z(G)} of G {\displaystyle G} is normal in G {\displaystyle G} . (In the special case that G {\displaystyle G} is abelian, the center is all of G {\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 N = { ( 1 ) , ( 123 ) , ( 132 ) } {\displaystyle N=\{(1),(123),(132)\}} of the symmetric group S 3 , {\displaystyle S_{3},} consisting of the identity and both three-cycles. In particular, one can check that every coset of N {\displaystyle N} is either equal to N {\displaystyle N} itself or is equal to ( 12 ) N = { ( 12 ) , ( 23 ) , ( 13 ) } . {\displaystyle (12)N=\{(12),(23),(13)\}.} On the other hand, the subgroup H = { ( 1 ) , ( 12 ) } {\displaystyle H=\{(1),(12)\}} is not normal in S 3 {\displaystyle S_{3}} since ( 123 ) H = { ( 123 ) , ( 13 ) } ≠ { ( 123 ) , ( 23 ) } = H ( 123 ) . {\displaystyle (123)H=\{(123),(13)\}\neq \{(123),(23)\}=H(123).} ^[11] This illustrates the general fact that any subgroup H ≤ G {\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 G L n ( R ) {\displaystyle \mathrm {GL} _{n}(\mathbf {R} )} of all invertible n × n {\displaystyle n\times n} matrices with real entries under the operation of matrix multiplication and its subgroup S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )} of all n × n {\displaystyle n\times n} matrices of determinant 1 (the special linear group). To see why the subgroup S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )} is normal in G L n ( R ) {\displaystyle \mathrm {GL} _{n}(\mathbf {R} )} , consider any matrix X {\displaystyle X} in S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )} and any invertible matrix A {\displaystyle A} . Then using the two important identities det ( A B ) = det ( A ) det ( B ) {\displaystyle \det(AB)=\det(A)\det(B)} and det ( A − 1 ) = det ( A ) − 1 {\displaystyle \det(A^{-1})=\det(A)^{-1}} , one has that det ( A X A − 1 ) = det ( A ) det ( X ) det ( A ) − 1 = det ( X ) = 1 {\displaystyle \det(AXA^{-1})=\det(A)\det(X)\det(A)^{-1}=\det(X)=1} , and so A X A − 1 ∈ S L n ( R ) {\displaystyle AXA^{-1}\in \mathrm {SL} _{n}(\mathbf {R} )} as well. This means S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )} is closed under conjugation in G L n ( R ) {\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 H {\displaystyle H} is a normal subgroup of G , {\displaystyle G,} and K {\displaystyle K} is a subgroup of G {\displaystyle G} containing H , {\displaystyle H,} then H {\displaystyle H} is a normal subgroup of K . {\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 G {\displaystyle G} and H {\displaystyle H} are normal subgroups of their direct product G × H . {\displaystyle G\times H.}
• If the group G {\displaystyle G} is a semidirect product G = N ⋊ H , {\displaystyle G=N\rtimes H,} then N {\displaystyle N} is normal in G , {\displaystyle G,} though H {\displaystyle H} need not be normal in G . {\displaystyle G.}
• If M {\displaystyle M} and N {\displaystyle N} are normal subgroups of an additive group G {\displaystyle G} such that G = M + N {\displaystyle G=M+N} and M ∩ N = { 0 } {\displaystyle M\cap N=\{0\}} , then G = M ⊕ N . {\displaystyle G=M\oplus N.} ^[18]
• Normality is preserved under surjective homomorphisms;^[19] that is, if G → H {\displaystyle G\to H} is a surjective group homomorphism and N {\displaystyle N} is normal in G , {\displaystyle G,} then the image f ( N ) {\displaystyle f(N)} is normal in H . {\displaystyle H.}
• Normality is preserved by taking inverse images;^[19] that is, if G → H {\displaystyle G\to H} is a group homomorphism and N {\displaystyle N} is normal in H , {\displaystyle H,} then the inverse image f − 1 ( N ) {\displaystyle f^{-1}(N)} is normal in G . {\displaystyle G.}
• Normality is preserved on taking direct products;^[20] that is, if N 1 ◃ G 1 {\displaystyle N_{1}\triangleleft G_{1}} and N 2 ◃ G 2 , {\displaystyle N_{2}\triangleleft G_{2},} then N 1 × N 2 ◃ G 1 × G 2 . {\displaystyle N_{1}\times N_{2}\;\triangleleft \;G_{1}\times G_{2}.}
• Every subgroup of index 2 is normal. More generally, a subgroup, H , {\displaystyle H,} of finite index, n , {\displaystyle n,} in G {\displaystyle G} contains a subgroup, K , {\displaystyle K,} normal in G {\displaystyle G} and of index dividing n ! {\displaystyle n!} called the normal core. In particular, if p {\displaystyle p} is the smallest prime dividing the order of G , {\displaystyle G,} then every subgroup of index p {\displaystyle p} is normal.^[21]
• The fact that normal subgroups of G {\displaystyle G} are precisely the kernels of group homomorphisms defined on G {\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.

Normal subgroups, quotient groups and homomorphisms

If N {\displaystyle 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 . {\displaystyle \left(a_{1}N\right)\left(a_{2}N\right):=\left(a_{1}a_{2}\right)N.} This relation defines a mapping G / N × G / N → G / N . {\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 a 1 , a 2 {\displaystyle a_{1},a_{2}} does not affect the result. To this end, consider some other representative elements a 1 ′ ∈ a 1 N , a 2 ′ ∈ a 2 N . {\displaystyle a_{1}'\in a_{1}N,a_{2}'\in a_{2}N.} Then there are n 1 , n 2 ∈ N {\displaystyle n_{1},n_{2}\in N} such that a 1 ′ = a 1 n 1 , a 2 ′ = a 2 n 2 . {\displaystyle 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 , {\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 N {\displaystyle N} is a normal subgroup, and therefore there is n 1 ′ ∈ N {\displaystyle n_{1}'\in N} such that n 1 a 2 = a 2 n 1 ′ . {\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 G / N . {\displaystyle G/N.} There is a natural homomorphism, f : G → G / N , {\displaystyle f:G\to G/N,} given by f ( a ) = a N . {\displaystyle f(a)=aN.} This homomorphism maps N {\displaystyle N} into the identity element of G / N , {\displaystyle G/N,} which is the coset e N = N , {\displaystyle eN=N,} ^[23] that is, ker ⁡ ( f ) = N . {\displaystyle \ker(f)=N.}

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

See also

Notes

References

Bibliography

Further reading

• I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, 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 in Abstract Algebra. The Basic Graduate Year
• Timothy Gowers, Normal subgroups and quotient groups
• John Baez, What's a Normal Subgroup?