Editing Quotient group (section)

== Examples ==
=== Even and odd integers ===
Consider the group of [[integer]]s <math>\Z</math> (under addition) and the subgroup <math>2\Z</math> consisting of all even integers. This is a normal subgroup, because <math>\Z</math> is [[abelian group|abelian]]. There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group <math>\Z\,/\,2\Z</math> is the cyclic group with two elements. This quotient group is isomorphic with the set <math>\left\{0,1 \right\}</math> with addition modulo&nbsp;2; informally, it is sometimes said that <math>\Z\,/\,2\Z</math> ''equals'' the set <math>\left\{0,1 \right\}</math> with addition modulo 2.

'''Example further explained...'''
: Let <math> \gamma(m) </math> be the remainders of <math> m \in \Z </math> when dividing by {{tmath|1= 2 }}. Then, <math> \gamma(m)=0 </math> when <math> m </math> is even and <math> \gamma(m)=1 </math> when <math> m </math> is odd.
: By definition of {{tmath|1= \gamma }},  the kernel of {{tmath|1= \gamma }}, {{tmath|1= \ker(\gamma) = \{ m \in \Z : \gamma(m)=0 \} }},  is the set of all even integers.
: Let {{tmath|1= H = \ker(\gamma) }}. Then, <math> H </math> is a subgroup, because the identity in {{tmath|1= \Z }}, which is {{tmath|1= 0 }}, is in {{tmath|1= H }}, the sum of two even integers is even and hence if <math> m </math> and <math> n </math> are in {{tmath|1= H }}, <math> m+n </math> is in <math> H </math> (closure) and if <math> m </math> is even, <math> -m </math> is also even and so <math> H </math> contains its inverses.
: Define <math> \mu : \mathbb{Z} / H \to \mathrm{Z}_2 </math> as <math> \mu(aH)=\gamma(a) </math> for <math> a\in\Z </math> and <math>\mathbb{Z} / H</math> is the quotient group of left cosets; {{tmath|1= \mathbb{Z} / H=\{H,1+H\} }}.
: Note that we have defined {{tmath|1= \mu }}, <math> \mu(aH) </math> is <math> 1 </math> if <math> a </math> is odd and <math> 0 </math> if <math> a </math> is even.
: Thus, <math> \mu </math> is an isomorphism from <math>\mathbb{Z} / H</math> to {{tmath|1= \mathrm{Z}_2 }}.

=== Remainders of integer division ===
A slight generalization of the last example. Once again consider the group of integers <math>\Z</math> under addition.  Let {{tmath|1= n }} be any positive integer. We will consider the subgroup <math>n\Z</math> of <math>\Z</math> consisting of all multiples of {{tmath|1= n }}. Once again <math>n\Z</math> is normal in <math>\Z</math> because <math>\Z</math> is abelian. The cosets are the collection {{tmath|1= \left\{n\Z, 1+n\Z, \; \ldots, (n-2)+n\Z, (n-1)+n\Z \right\} }}. An integer <math>k</math> belongs to the coset {{tmath|1= r+n\Z }}, where <math>r</math> is the remainder when dividing <math>k</math> by {{tmath|1= n }}. The quotient <math>\Z\,/\,n\Z</math> can be thought of as the group of "remainders" modulo {{tmath|1= n }}. This is a [[cyclic group]] of order {{tmath|1= n }}.

=== Complex integer roots of 1 ===
[[File:Normal subgroup illustration.svg|right|thumb|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 number|complex]] [[unit circle]], form a multiplicative abelian group {{tmath|1= G }}, shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup <math>N</math> 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 <math>G\,/\,N</math> is the group of three colors, which turns out to be the cyclic group with three elements.

=== Real numbers modulo the integers ===
Consider the group of [[real number]]s <math>\R</math> under addition, and the subgroup <math>\Z</math> of integers. Each coset of <math>\Z</math> in <math>\R</math> is a set of the form {{tmath|1= a+\Z }}, where <math>a</math> is a real number. Since <math>a_1+\Z</math> and <math>a_2+\Z</math> are identical sets when the non-[[integer part]]s of <math>a_1</math> and <math>a_2</math> are equal, one may impose the restriction <math>0 \leq a < 1</math> 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 <math>\R\,/\,\Z</math> is isomorphic to the [[circle group]], the group of [[complex number]]s of [[absolute value]] 1 under multiplication, or correspondingly, the group of [[rotation]]s in 2D about the origin, that is, the special [[orthogonal group]] {{tmath|1= \mathrm{SO}(2) }}. An isomorphism is given by <math>f(a+\Z) = \exp(2\pi ia)</math> (see [[Euler's identity]]).

=== Matrices of real numbers ===
If <math>G</math> is the group of invertible <math>3 \times 3</math> real [[matrix (mathematics)|matrices]], and <math>N</math> is the subgroup of <math>3 \times 3</math> real matrices with [[determinant]] 1, then <math>N</math> is normal in <math>G</math> (since it is the [[kernel (algebra)|kernel]] of the determinant [[group homomorphism|homomorphism]]). The cosets of <math>N</math> are the sets of matrices with a given determinant, and hence <math>G\,/\,N</math> is isomorphic to the multiplicative group of non-zero real numbers. The group <math>N</math> is known as the [[special linear group]] {{tmath|1= \mathrm{SL}(3) }}.

=== Integer modular arithmetic ===
Consider the abelian group <math>\mathrm{Z}_4 = \Z\,/\,4 \Z</math> (that is, the set <math>\left\{0, 1, 2, 3 \right\}</math> with addition [[Modular arithmetic|modulo]] 4), and its subgroup {{tmath|1= \left\{0, 2\right\} }}. The quotient group <math>\mathrm{Z}_4\,/\,\left\{0, 2\right\}</math> is {{tmath|1= \left\{\left\{ 0, 2 \right\}, \left\{1, 3 \right\} \right\} }}. This is a group with identity element {{tmath|1= \left\{0, 2\right\} }}, and group operations such as {{tmath|1= \left\{0, 2 \right\} + \left\{1, 3 \right\} = \left\{1, 3 \right\} }}.  Both the subgroup <math>\left\{0, 2\right\}</math> and the quotient group <math>\left\{\left\{ 0, 2 \right\}, \left\{1, 3 \right\} \right\}</math> are isomorphic with {{tmath|1= \mathrm{Z}_2 }}.

=== Integer multiplication ===
Consider the multiplicative group {{tmath|1= G=(\Z_{n^2})^{\times} }}. The set <math>N</math> of {{tmath|1= n }}th residues is a multiplicative subgroup isomorphic to {{tmath|1= (\Z_{n})^{\times} }}. Then <math>N</math> is normal in <math>G</math> and the factor group <math>G\,/\,N</math> has the cosets {{tmath|1= N, (1+n)N, (1+n)2N, \;\ldots, (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 <math>G</math> without knowing the factorization of {{tmath|1= n }}.