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abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
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^ \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.


Definitions

A
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
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. The usual notation for this relation is N \triangleleft G.


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. * The image of conjugation of N by any element of G is a subset of N, i.e., gNg^\subseteq N for all g\in G. * The image of conjugation of N by any element of G is equal to N, i.e., gNg^= N for all g\in G. * For all g \in G, the left and right cosets gN and Ng are equal. * The sets of left and right cosets of N in G coincide. * Multiplication in G preserves the equivalence relation "is in the same left coset as". That is, for every g,g',h,h'\in G satisfying g N = g' N and h N = h' N, we have (g h) N = (g' h') N. * There exists a group on the set of left cosets of N where multiplication of any two left cosets gN and hN yields the left coset (gh)N. (This group is called the ''quotient group'' of G ''modulo'' N, denoted G/N.) * N is a union of conjugacy classes of G. * N is preserved by the inner automorphisms of G. * There is some group homomorphism G \to H whose kernel is N. * There exists a group homomorphism \phi:G \to H whose
fibers Fiber (spelled fibre in British English; from ) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often inco ...
form a group where the identity element is N and multiplication of any two fibers \phi^(h_1) and \phi^(h_2) yields the fiber \phi^(h_1 h_2). (This group is the same group G/N mentioned above.) * There is some congruence relation on G for which the equivalence class of the
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
is N. * For all n\in N and g\in G, the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
,g= n^ g^ n g is in N. * Any two elements commute modulo the normal subgroup membership relation. That is, for all g, h \in G, g h \in N if and only if h g \in N.


Examples

For any group G, the trivial subgroup \ consisting of just the identity element of G is always a normal subgroup of G. Likewise, G itself is always a normal subgroup of G. (If these are the only normal subgroups, then G is said to be
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...
.) 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 More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup. If G is an abelian group then every subgroup N of G is normal, because gN = \_ = \_ = Ng. More generally, for any group G, every subgroup of the ''center'' Z(G) of G is normal in G. (In the special case that G is abelian, the center is all of 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. A concrete example of a normal subgroup is the subgroup N = \ of the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
S_3, consisting of the identity and both three-cycles. In particular, one can check that every coset of N is either equal to N itself or is equal to (12)N = \. On the other hand, the subgroup H = \ is not normal in S_3 since (123)H = \ \neq \ = H(123). This illustrates the general fact that any subgroup H \leq G of index two is normal. As an example of a normal subgroup within a matrix group, consider the
general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
\mathrm_n(\mathbf) of all invertible n\times n matrices with real entries under the operation of matrix multiplication and its subgroup \mathrm_n(\mathbf) of all n\times n matrices of
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
1 (the special linear group). To see why the subgroup \mathrm_n(\mathbf) is normal in \mathrm_n(\mathbf), consider any matrix X in \mathrm_n(\mathbf) and any invertible matrix A. Then using the two important identities \det(AB)=\det(A)\det(B) and \det(A^)=\det(A)^, one has that \det(AXA^) = \det(A) \det(X) \det(A)^ = \det(X) = 1, and so AXA^ \in \mathrm_n(\mathbf) as well. This means \mathrm_n(\mathbf) is closed under conjugation in \mathrm_n(\mathbf), so it is a normal subgroup. 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. The translation group is a normal subgroup of the Euclidean group in any dimension. 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 is a normal subgroup of G, and K is a subgroup of G containing H, then H is a normal subgroup of K. * 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 In mathematics, a binary relation on a set (mathematics), set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Every partial order and every equivalence relation is transitive. For example ...
. The smallest group exhibiting this phenomenon is the
dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
of order 8. However, a characteristic subgroup of a normal subgroup is normal. A group in which normality is transitive is called a T-group. * 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. * If M and N are normal subgroups of an additive group G such that G = M + N and M \cap N = \, then G = M \oplus N. * Normality is preserved under surjective homomorphisms; that is, 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; that is, if G \to H is a group homomorphism and N is normal in H, then the inverse image f^(N) is normal in G. * Normality is preserved on taking direct products; that is, 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 Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...
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. * 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 Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...
if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime
index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...
, 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 Mand their product N M = \ are also normal subgroups of G. The normal subgroups of G form a lattice under subset inclusion with least element, \, and
greatest element In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined duality (order theory), dually ...
, 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.


Normal subgroups, quotient groups and homomorphisms

If N is a normal subgroup, we can define a multiplication on cosets as follows: \left(a_1 N\right) \left(a_2 N\right) := \left(a_1 a_2\right) 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 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) = a N. This homomorphism maps N into the identity element of G/N, which is the coset e N = N, 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 \ 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). 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). 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.


See also


Operations taking subgroups to subgroups

* Normalizer * Normal closure * Normal core


Subgroup properties complementary (or opposite) to normality

* Malnormal subgroup * Contranormal subgroup * Abnormal subgroup * Self-normalizing subgroup


Subgroup properties stronger than normality

* Characteristic subgroup * Fully characteristic subgroup


Subgroup properties weaker than normality

* Subnormal subgroup * Ascendant subgroup * Descendant subgroup * Quasinormal subgroup * Seminormal subgroup * Conjugate permutable subgroup * Modular subgroup * Pronormal subgroup * Paranormal subgroup * Polynormal subgroup * C-normal subgroup


Related notions in algebra

* Ideal (ring theory) * Semigroup ideal


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

* {{MathWorld, urlname=NormalSubgroup, title= normal subgroup
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


Subgroup properties