# Quotient group

In

mathematics , given a group "G" and anormal 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 theidentity element . The quotient group is written "G"/"N" and is usually spoken in English as "G" mod "N" ("mod" is short formodulo ). 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 ahomogeneous space .**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 "ST" = { "st" : "s" in "S" and "t" in "T" }. This operation is

associative and has asidentity element thesingleton {"e"}, where "e" is the identity element of "G". Thus, the set of all subsets of "G" forms amonoid 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 thecoset s 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 "N" ⊲ "G". A subgroup that permutes with every subgroup of "G" is called apermutable subgroup .**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., "G"/"N" = { "aN" : "a" in "G" }. 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"). This operation is closed, because ("aN")("bN") really is a left coset::("aN")("bN") = "a"("Nb")"N" = "a"("bN")"N" = ("ab")"NN" = ("ab")"N".

The normality of "N" is used in this equation. Because of the normality of "N", the left cosets and right cosets of "N" in "G" are equal, and so "G"/"N" could be defined as the set of right cosets of "N" in "G". 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"^{−1}"N".**Motivation for definition**The reason "G"/"N" is called a quotient group comes from

division ofinteger s. 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 than 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

integer s**Z**(under addition) and the subgroup 2**Z**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**/2**Z**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**/2**Z**"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 "n**"Z**of**Z**consisting of all multiples of "n". Once again "n**"Z**is normal in**Z**because**Z**is abelian. The cosets are the collection {"n**"Z**,1+"n**"Z**,...,("n"−2)+"n**"Z**,("n"−1)+"n**"Z**}. An integer "k" belongs to the coset "r"+"n**"Z**, where "r" is the remainder when dividing "k" by "n". The quotient**Z**/"n**"Z**can be thought of as the group of "remainders" modulo "n". This is acyclic group of order "n".*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 number s**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 thecircle group S^{1}, the group ofcomplex number s ofabsolute value 1 under multiplication, or correspondingly, the group ofrotation s in 2D about the origin, i.e., the specialorthogonal group SO(2). An isomorphism is given by "f"("a" +**Z**) = exp(2"πia") (seeEuler'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

**Z**_{4}=**Z**/4**Z**(that is, the set { 0, 1, 2, 3 } with addition modulo 4), and its subgroup { 0, 2 }. The quotient group**Z**_{4}/ { 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**Z**_{2}.*Consider the multiplicative group $G=mathbf\{Z\}^*\_\{n^2\}$. The set "N" of "n"th residues is a multiplicative subgroup of order "ϕ"("n") of $mathbf\{Z\}^*\_n$. Then "N" is normal in "G" and the factor group "G"/"N" has the cosets "N", (1+"n")"N", (1+"n")

^{2}N,…,(1+"n")^{"n"−1}N. ThePallier cryptosystem is based on theconjecture 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" 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**"/" 2**Z**).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 theisomorphism theorem s.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 contained in the center of "G", 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 "C"

_{2}. 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 orsemidirect product . The problem of determining when this is the case is known as theextension problem . An example where it is "not" possible is as follows.**Z**_{4}/ { 0, 2 } is isomorphic to**Z**_{2}, and { 0, 2 } also, but the only semidirect product is the direct product, because**Z**_{2}has only the trivialautomorphism . Therefore**Z**_{4}, which is different from**Z**_{2}×**Z**_{2}, cannot be reconstructed.**ee also***

Quotient ring , also called a "factor ring"

*Group extension

*Extension problem

*Lattice theorem

*Quotient category

*Short exact sequence

*Wikimedia Foundation.
2010.*

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