Normal subgroup

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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 ${\displaystyle N}$ 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]

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 ${\displaystyle N\triangleleft G}$.

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]}

For any group ${\displaystyle G}$, the trivial subgroup ${\displaystyle \{e\}}$ consisting of only 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.

• 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.

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]

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 ${\displaystyle G}$, ${\displaystyle G/N}$, 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]

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

• 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?