Normal Subgroups and Factor Groups

Normal Subgroups

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate 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 $N$ of the group $G$ is normal in $G$ if and only if $gng^{-1}\in N$ 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 $N$ 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 $N.$
• The image of conjugation of $N$ by any element of $G$ is equal to $N.$
• 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.
• 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 $gh$: for all $x,y,g,h\in G,$ if $x\in gN$and $y\in hN$ then $xy\in (gh)N.$
• $N$ 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 $N.$
• For all $n\in N$ and $g\in G,$ the commutator $[n,g]=n^{-1}g^{-1}ng$ is in $N.$
• Any two elements commute regarding the normal subgroup membership relation: for all $g,h\in G,$ $gh\in N$ if and only if $hg\in N.$

Examples

For any group $G,$ the trivial subgroup $\{e\}$ 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.) 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 $[G,G].$ 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 $gN=\{gn\}_{n\in N}=\{ng\}_{n\in N}=Ng.$ 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 $N=\{(1),(123),(132)\}$ of the symmetric group $S_{3},$ 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 $(12)N=\{(12),(23),(13)\}.$ On the other hand, the subgroup $H=\{(1),(12)\}$ is not normal in $S_{3}$ since $(123)H=\{(123),(13)\}\neq \{(123),(23)\}=H(123).$ This illustrates the general fact that any subgroup $H\leq G$ of index two is normal.

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. The smallest group exhibiting this phenomenon is the dihedral group 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 $G.$
• 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 $f^{-1}(N)$ 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 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 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, 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 $N\cap M$and their product $NM=\{nm:n\in N\;{\text{ and }}\;m\in M\}$ are also normal subgroups of $G.$

The normal subgroups of $G$ form a lattice under subset inclusion with least element, $\{e\},$ and greatest element, $G.$ 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 normal 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 $f(a)=aN.$ This homomorphism maps $N$ into the identity element of $G/N,$ which is the coset $eN=N,$ 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 $\{e\}$ 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). 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 $G.$

Factor Groups

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory.

In a quotient of a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written G / N, where G is the original group and N is the normal subgroup. (This is pronounced "G mod N", where "mod" is short for modulo.)

Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of G. Specifically, the image of G under a homomorphism φ: G → H is isomorphic to G / ker(φ) where ker(φ) denotes the kernel of φ.

The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects.

Definition and illustration

Given a group G and a subgroup H, and an element a ∈ G, one can consider the corresponding left coset: aH := { ah : h ∈ H }. Cosets are a natural class of subsets of a group; for example consider the abelian group G of integers, with operation defined by the usual addition, and the subgroup H of even integers. Then there are exactly two cosets: 0 + H, which are the even integers, and 1 + H, which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation).

For a general subgroup H, it is desirable to define a compatible group operation on the set of all possible cosets, { aH : a ∈ G }. This is possible exactly when H is a normal subgroup, see below. A subgroup N of a group G is normal if and only if the coset equality aN = Na holds for all a ∈ G. A normal subgroup of G is denoted N ◁ G.

Definition

Let N be a normal subgroup of a group G. Define the set G/N to be the set of all left cosets of N in G. That is, G/N = {aN : a ∈ G}. Since the identity element e ∈ N, a ∈ aN. Define a binary operation on the set of cosets, G/N, as follows. For each aN and bN in G/N, the product of aN and bN, (aN)(bN), is (ab)N. This works only because (ab)N does not depend on the choice of the representatives, a and b, of each left coset, aN and bN. To prove this, suppose xN = aN and yN = bN for some x, y, a, b ∈ G. Then

(ab)N = a(bN) = a(yN) = a(Ny) = (aN)y = (xN)y = x(Ny) = x(yN) = (xy)N.

This depends on the fact that N is a normal subgroup. It still remains to be shown that this condition is not only sufficient but necessary to define the operation on G/N.

To show that it is necessary, consider that for a subgroup N of G, we have been given that the operation is well defined. That is, for all xN = aN and yN = bN, for x, y, a, b ∈ G, (ab)N = (xy)N.

Let n ∈ N and g ∈ G. Since eN = nN, we have, gN = (eg)N = (ng)N.

Now, gN = (ng)N ⇔ N = g⁻¹(ng)N ⇔ g⁻¹ng ∈ N ∀ n ∈ N and g ∈ G.

Hence N is a normal subgroup of G.

It can also be checked that this operation on G/N is always associative. G/N has identity element N and the inverse of element aN can always be represented by a⁻¹N. Therefore, the set G/N together with the operation defined by (aN)(bN) = (ab)N forms a group, the quotient group of G by N.

Due to the normality of N, the left cosets and right cosets of N in G are the same, and so, G/N could have been defined to be the set of right cosets of N in G.

Example: Addition modulo 6

For example, consider the group with addition modulo 6: G = {0, 1, 2, 3, 4, 5}. Consider the subgroup N = {0, 3}, which is normal because G is abelian. Then the set of (left) cosets is of size three:

G/N = { a+N : a ∈ G } = { {0, 3}, {1, 4}, {2, 5} } = { 0+N, 1+N, 2+N }.

The binary operation defined above makes this set into a group, known as the quotient group, which in this case is isomorphic to the cyclic group of order 3.

Motivation for the name "quotient"

The reason G/N is called a quotient group comes from division of integers. When dividing 12 by 3 one obtains the answer 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, although we end up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects.

To elaborate, when looking at G/N with N a normal subgroup of G, the group structure is used to form a natural "regrouping". These are the cosets of N in G. Because we started with a group and normal subgroup, the final quotient contains more information than just the number of cosets (which is what regular division yields), but instead has a group structure itself.

Examples

Even and odd integers

Consider the group of integers Z (under addition) and the subgroup 2Z consisting of all even integers. This is a normal subgroup, because Z is abelian. There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group Z/2Z is the cyclic group with two elements. This quotient group is isomorphic with the set {0,1} with addition modulo 2; informally, it is sometimes said that Z/2Z equals the set {0,1} with addition modulo 2.

Example further explained...

Let $\gamma (m)=$ remainders of $m\in \mathbb {Z}$ when dividing by $2$.

Then $\gamma (m)=0$ when $m$ is even and $\gamma (m)=1$ when $m$ is odd.

By definition of $\gamma$, the kernel of $\gamma$,

ker($\gamma$) $=\{m\in \mathbb {Z} :\gamma (m)=0\}$, is the set of all even integers.

Let $H=$ ker($\gamma$).

Then $H$ is a subgroup, because the identity in $\mathbb {Z}$, which is $0$, is in $H$,

the sum of two even integers is even and hence if $m$ and $n$ are in $H$, $m+n$ is in $H$ (closure)

and if $m$ is even, $-m$ is also even and so $H$ contains its inverses.

Define $\mu :$ $\mathbb {Z}$ / H$\to \mathbb {Z} _{2}$ as $\mu (aH)=\gamma (a)$ for $a\in \mathbb {Z}$

and $\mathbb {Z}$ / H is the quotient group of left cosets; $\mathbb {Z}$ / H$=\{H,1+H\}$.

By the way we have defined $\mu$, $\mu (aH)$ is $1$ if $a$ is odd and $0$ if $a$ is even.

Thus, $\mu$ is an isomorphism from $\mathbb {Z}$ / H to $\mathbb {Z} _{2}$.

Remainders of integer division

A slight generalization of the last example. Once again consider the group of integers Z under addition. Let n be any positive integer. We will consider the subgroup nZ of Z consisting of all multiples of n. Once again nZ is normal in Z because Z is abelian. The cosets are the collection {nZ, 1+nZ, ..., (n−2)+nZ, (n−1)+nZ}. An integer k belongs to the coset r+nZ, where r is the remainder when dividing k by n. The quotient Z/nZ can be thought of as the group of "remainders" modulo n. This is a cyclic group of order n.

Complex integer roots of 1

The cosets of the fourth roots of unity N in the twelfth roots of unity G.

The twelfth roots of unity, which are points on the complex unit circle, form a multiplicative abelian group G, shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup N made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group into three cosets, shown in red, green and blue. One can check that the cosets form a group of three elements (the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.). Thus, the quotient group G/N is the group of three colors, which turns out to be the cyclic group with three elements.

The real numbers modulo the integers

Consider the group of real numbers R under addition, and the subgroup Z of integers. Each coset of Z in R is a set of the form a+Z, where a is a real number. Since a₁+Z and a₂+Z are identical sets when the non-integer parts of a₁ and a₂ are equal, one may impose the restriction Template:Nowrap without change of meaning. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. The quotient group R/Z is isomorphic to the circle group, the group of complex numbers of absolute value 1 under multiplication, or correspondingly, the group of rotations in 2D about the origin, that is, the special orthogonal group SO(2). An isomorphism is given by f(a+Z) = exp(2πia) (see Euler's identity).

Matrices of real numbers

If G is the group of invertible 3 × 3 real matrices, and N is the subgroup of 3 × 3 real matrices with determinant 1, then N is normal in G (since it is the kernel of the determinant homomorphism). The cosets of N are the sets of matrices with a given determinant, and hence G/N is isomorphic to the multiplicative group of non-zero real numbers. The group N is known as the special linear group SL(3).

Integer modular arithmetic

Consider the abelian group Z₄ = Z/4Z (that is, the set { 0, 1, 2, 3 } with addition modulo 4), and its subgroup { 0, 2 }. The quotient group Z₄/{ 0, 2 } is { { 0, 2 }, { 1, 3 } }. This is a group with identity element { 0, 2 }, and group operations such as { 0, 2 } + { 1, 3 } = { 1, 3 }. Both the subgroup { 0, 2 } and the quotient group { { 0, 2 }, { 1, 3 } } are isomorphic with Z₂.

Integer multiplication

Consider the multiplicative group $G=\mathbf {Z} _{n^{2}}^{*}$. The set N of nth residues is a multiplicative subgroup isomorphic to $\mathbf {Z} _{n}^{*}$. Then N is normal in G and the factor group G/N has the cosets N, (1+n)N, (1+n)²N, ..., (1+n)ⁿ⁻¹N. The Paillier cryptosystem is based on the conjecture that it is difficult to determine the coset of a random element of G without knowing the factorization of n.

Properties

The quotient group G/G is isomorphic to the trivial group (the group with one element), and G/{e} is isomorphic to G.

The order of G/N, by definition the number of elements, is equal to |G : N|, the index of N in G. If G is finite, the index is also equal to the order of G divided by the order of N. The set G/N may be finite, although both G and N are infinite (for example, Z/2Z).

There is a "natural" surjective group homomorphism π : G → G/N, sending each element g of G to the coset of N to which g belongs, that is: π(g) = gN. The mapping π is sometimes called the canonical projection of G onto G/N. Its kernel is N.

There is a bijective correspondence between the subgroups of G that contain N and the subgroups of G/N; if H is a subgroup of G containing N, then the corresponding subgroup of G/N is π(H). This correspondence holds for normal subgroups of G and G/N as well, and is formalized in the lattice theorem.

Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.

If G is abelian, nilpotent, solvable, cyclic or finitely generated, then so is G/N.

If H is a subgroup in a finite group G, and the order of H is one half of the order of G, then H is guaranteed to be a normal subgroup, so G/H exists and is isomorphic to C₂. This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if p is the smallest prime number dividing the order of a finite group, G, then if G/H has order p, H must be a normal subgroup of G.

Given G and a normal subgroup N, then G is a group extension of G/N by N. One could ask whether this extension is trivial or split; in other words, one could ask whether G is a direct product or semidirect product of N and G/N. This is a special case of the extension problem. An example where the extension is not split is as follows: Let G = Z₄ = {0, 1, 2, 3}, and N = {0, 2}, which is isomorphic to Z₂. Then G/N is also isomorphic to Z₂. But Z₂ has only the trivial automorphism, so the only semi-direct product of N and G/N is the direct product. Since Z₄ is different from Z₂ × Z₂, we conclude that G is not a semi-direct product of N and G/N.

Quotients of Lie groups

If $G$ is a Lie group and $N$ is a normal and closed (in the topological rather than the algebraic sense of the word) Lie subgroup of $G$, the quotient $G$ / $N$ is also a Lie group. In this case, the original group $G$ has the structure of a fiber bundle (specifically, a principal $N$-bundle), with base space $G$ / $N$ and fiber $N$. The dimension of $G$ / $N$ equals $\mathrm {dim} \ G-\mathrm {dim} \ N$.

Note that the condition that $N$ is closed is necessary. Indeed, if $N$ is not closed then the quotient space is not a T1-space (since there is a coset in the quotient which cannot be separated from the identity by an open set), and thus not a Hausdorff space.

For a non-normal Lie subgroup $N$, the space $G$ / $N$ of left cosets is not a group, but simply a differentiable manifold on which $G$ acts. The result is known as a homogeneous space.

Licensing

Content obtained and/or adapted from:

• Normal subgroup, Wikipedia under a CC BY-SA license
• Quotient group, Wikipedia under a CC BY-SA license