Quotient group

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In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that "collapses" the normal subgroup N to the identity element. The quotient group is written G/N and is usually spoken in English as G mod N (mod is short for modulo). If N is not a normal subgroup, a quotient may still be taken, but the result will not be a group; rather, it will be a homogeneous space.

1 The product of subsets of a group
2 Definition
3 Motivation for definition
4 Examples
5 Properties
6 See also

The product of subsets of a group

In the following discussion, we will use a binary operation on the subsets of G: if two subsets S and T of G are given, we define their product as:

This operation is associative and has as identity element the singleton {e}, where e is the identity element of G. Thus, the set of all subsets of G forms a monoid under this operation. In terms of this operation we can first explain what a quotient group is, and then explain what a normal subgroup is:

A quotient group of a group G is a partition of G which is itself a group under this operation.

It is fully determined by the subset containing e. A normal subgroup of G is the set containing e in any such partition. The subsets in the partition are the cosets of this normal subgroup. A subgroup N of a group G is normal if and only if the coset equality aN = Na holds for all a in G. In terms of the binary operation on subsets defined above, a normal subgroup of G is a subgroup that commutes with every subset of G and is denoted <math>N \triangleleft G</math>. A formal definition is <math>N \triangleleft G\,\,\mathrm{iff}\,\,\forall\,n\in{N},g\in{G}\ gng^{-1}\in{N}</math>.

Definition

Let N be a normal subgroup of a group G. We define the set G/N to be the set of all left cosets of N in G, i.e.,

The group operation on G/N is the product of subsets defined above. In other words, for each aN and bN in G/N, the product of aN and bN is (aN)(bN). For this operation to be closed, we must show that (aN)(bN) really is a left coset:

(aN)(bN) = a(Nb)N = a(bN)N = (ab)NN = (ab)N.

Note that we have already used the normality of N in this equation. Also note that because of the normality of N, we could have chosen to define G/N as the set of right cosets of N in G. Also note that because the operation is derived from the product of subsets of G, the operation is well-defined (does not depend on the particular choice of representatives), associative and has identity element N. The inverse of an element aN of G/N is a⁻¹N. This completes the proof that G/N is a group.

Motivation for definition

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, however we end up with a group for a final answer instead of a number because groups have more structure then a random 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

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; 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.
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}. For any integer k, it 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 also isomorphic to the cyclic group of order n.

The cosets of N in G

Consider the multiplicative abelian group G of complex twelfth roots of unity, which are points on the unit circle, 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.
Consider the group of real numbers R under addition, and the subgroup Z of integers. The cosets of Z in R are all sets of the form a + Z, with 0 ≤ a < 1 a real number. 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 S¹, the group of complex numbers of absolute value 1 under multiplication, or correspondingly, the group of rotations in 2D about the origin, i.e., the special orthogonal group SO(2). An isomorphism is given by f(a + Z) = exp(2πia) (see Euler's identity).
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.
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 } }, with their group operations induced by cyclic group Z₄, are isomorphic with Z₂.

Consider the multiplicative group <math>G=Z^*_{n^2}</math>. The set <math>N</math> of n-th residues is a multiplicative subgroup of order <math>\varphi(n)</math> of <math>Z^*_n</math>. Then <math>N</math> is normal in <math>G</math> and the factor group <math>G/N</math> has the cosets <math>N, (1+n)N, (1+n)^2 N,\dots,(1+n)^{n-1} N</math>. The Pallier cryptosystem is based on the conjecture that it is difficult to determine the coset of a random element of <math>G</math> without knowing the factorization of <math>n</math>.

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 is by definition 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. Note that G / N may be finite, although both G and N are infinite (e.g. 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 or solvable, then so is G / N. If G is cyclic or finitely generated, then so is G / N. If N is abelian, then G is called the central extension of the quotient group. 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. Every group is isomorphic to a quotient of a free group. Sometimes, but not necessarily, a group G can be reconstructed from G / N and N, as a direct product or semidirect product. The problem of determining when this is the case is known as the extension problem. An example where it is not possible is as follows. Z₄ / { 0, 2 } is isomorphic to Z₂, and { 0, 2 } also, but the only semidirect product is the direct product, because Z₂ has only the trivial automorphism. Therefore Z₄, which is different from Z₂ × Z₂, cannot be reconstructed.