Quotient group - Citizendia

category of groups
Basic notions in group theory
subgroups, normal subgroups
quotient groups
group homomorphisms, kernel, image
(semi-)direct product, direct sum
types of groups
finite, infinite
discrete, continuous
multiplicative, additive
abelian, cyclic, simple, solvable

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. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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 SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning 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 Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that The quotient group is written G/N and is usually spoken in English as G mod N (mod is short for modulo). The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure 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. In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group

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 ST = { st : s in S and t in T }. This operation is associative and has as identity element the singleton {e}, where e is the identity element of G. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that Thus, the set of all subsets of G forms a monoid under this operation. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary 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. In Mathematics, a partition of a set X is a division of X into non-overlapping " parts " or " blocks "

It is fully determined by the subset containing e. A normal subgroup of G is the set containing e in any such partition. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. The subsets in the partition are the cosets of this normal subgroup. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH

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 a permutable subgroup. In Mathematics, in the field of Group theory, a quasinormal subgroup, or permutable subgroup, is a Subgroup of a group that commutes

Definition

Let N be a normal subgroup of a group G. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. 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. In Mathematics, the term well-defined is used to specify that a certain concept or object (a function, a property, a relation, etc The inverse of an element aN of G/N is a⁻¹N.

Motivation for definition

The reason G/N is called a quotient group comes from division of integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French 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 integers Z (under addition) and the subgroup 2Z consisting of all even integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French This is a normal subgroup, because Z is abelian. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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}. 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. 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

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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power In Mathematics, a unit circle is 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. In Mathematics, the real numbers may be described informally in several different ways 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. In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation e. , the special orthogonal group SO(2). In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n An isomorphism is given by f(a + Z) = exp(2πia) (see Euler's identity). In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where

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). In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n 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 In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function 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 }. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers 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.

Consider the multiplicative group $G=\mathbf{Z}^*_{n^2}$ . The set N of nth 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)²N,…,(1+n)ⁿ⁻¹N. 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 Paillier cryptosystem, named after and invented by Pascal Paillier in 1999, is a probabilistic Asymmetric algorithm for Public key cryptography In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of

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 In Mathematics, the term trivial is frequently used for objects (for examples groups or Topological spaces that have a very simple

The order of G / N is by definition equal to [G : N], the index of N in G. In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH 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. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function The mapping π is sometimes called the canonical projection of G onto G / N. Its kernel is N. 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

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 improperly referred to as the fourth Isomorphism theorem or the correspondence theorem states that there exists a

Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems. In Abstract algebra, the Fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural

If G is abelian, nilpotent or solvable, then so is G / N. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the 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

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 In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the

If N is is contained in the center of G, then G is called the central extension of the quotient group. In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the

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. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be

Sometimes, but not necessarily, a group G can be reconstructed from G / N and N, as a direct product or semidirect product. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can 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. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself Therefore Z₄, which is different from Z₂ × Z₂, cannot be reconstructed.

Contents

The product of subsets of a group

Definition

Motivation for definition

Examples

Properties

See also