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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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
This picture illustrates how the hours on a clock form a group under modular addition. ...
In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...
The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ...
In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...
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 mathematics, associativity is a property that a binary operation can have. ...
For other uses, see identity (disambiguation). ...
Generally, a singleton is something which exists alone in some way. ...
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...
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 partition of U into 6 blocks: an Euler diagram representation. ...
In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...
In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...
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  . A formal definition is  .
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., In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...
 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. In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...
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. Look up division in Wiktionary, the free dictionary. ...
The integers are commonly denoted by the above symbol. ...
To elaborate, when looking at G/N with N a normal subgroup of G, the group structure is used to for 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 {0+Z,1+Z,...,(n-2)+Z,(n-1)+Z}. For any integer k, it belongs to the coset r+Z, 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.
- 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
 . The set N of n-th residues is a multiplicative subgroup of order  of  . Then N is normal in G and the factor group G / N has the cosets  . The Pallier 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. The integers are commonly denoted by the above symbol. ...
In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...
In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of...
Image File history File links Download high resolution version (648x623, 48 KB) Summary Made by myself with Matlab Licensing I, the creator of this work, hereby release it into the public domain. ...
Image File history File links Download high resolution version (648x623, 48 KB) Summary Made by myself with Matlab Licensing I, the creator of this work, hereby release it into the public domain. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, the nth roots of unity, or de Moivre numbers, are all the complex numbers which yield 1 when raised to a given power n. ...
Illustration of a unit circle. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...
This article is about rotation as a movement of a physical body. ...
In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
For other uses, see List of topics named after Leonhard Euler. ...
In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ...
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...
In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of...
The Paillier cryptosystem is a probabilistic asymmetric algorithm for public key cryptography, invented by Pascal Paillier in 1999. ...
In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ...
Properties The quotient group G / G is isomorphic to the trivial group (the group with one element), and G / {e} is isomorphic to G. In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...
In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. ...
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). It has been suggested that this article or section be merged with multiplicative order. ...
In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...
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. In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ...
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. In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem, states that there exists a bijection from the set of all subgroups of a group G containing a normal subgroup N onto the set of all subgroups of the quotient group G/N. This means that the...
Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems. In abstract algebra, for a number of algebraic structures, the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism. ...
In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ...
If G is abelian, nilpotent or solvable, then so is G / N. In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...
In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ...
In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. ...
If G is cyclic or finitely generated, then so is G / N. In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense that the group has an element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of...
In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...
If N is abelian, then G is called the central extension of the quotient group. In group theory, a central extension of a group G is an exact sequence of groups such that A is in Z(E), the center of the group E. Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to...
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. The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many...
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. In mathematics, one can often define a direct product of objects already known, giving a new one. ...
In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, one of which is normal. ...
In group theory, if the factor group G/K is isomorphic to H, one says that G is an extension of H by K. To consider some examples, if G = H × K, then G is an extension of both H and K. More generally, if G is a semidirect product...
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...
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